TRANS L ATI O N
NASA U f-542
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AERODYNAMICS AND FLIGHT DYNAMICS
OF TURBOJET AIRCRAFT
by T. I. Ligmn
Transport Press, Moscow, 1967
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION • WASHINGTON, D. C. • SEPTEMBER 1969
TECH LIBRARY KAFB, NM
iN-fXlOfX X J. x-ua.«j
AERODYNAMICS AND FLIGHT DYNAMICS OF TURBOJET AIRCRAFT
By T. I. Ligum
Translation of "Aerodinamika i Dinamika Poleta
Transport Press, Moscow, 1967
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
For sal* by the Clearinghouse for Federal Scientific and Technical Information
Springfield, Virginia 22151 - CFSTI price $3.00
Table of Contents
Chapter I. The Physical Basis of High-Speed Aerodynamics
§1. Variations in the Parameters of Air with Altitude.
The Standard Atmosphere ......
§2. Compressibility of Air. ......
§3. The Propagation of Small Disturbences in Air
Sound and Sound Waves .......
§^. The Speed of Sound as a Criterion for the Compressibility
of Gases .........
§5. The Mach Number and its Value in Flight Problems
§6. Flight Speed. Corrections to Instrument Readings Necessitated
by Compressibility .......
§7. The Character of the Propagation of Minor Perturbations
in Flight at Various Altitudes .....
§8. Trans- or Supersonic Flow of Air Around Bodies .
§9. Sonic "boom". ........
§10. Features of the Formation of Compression Shock During Flow
Around Various Shapes of Bodies. ....
§11. Critical Mach Number. The Effect of Compressibility on the
Motion of Air Flying Around a Wing ....
§12. The Dependence of the Speed of the Gas Flow on the Shape
of the Channel. The Laval Nozzle ....
§13. Laminar and Turbulent Flow of Ai r
§1^. Pressure Distribution at Sub- and Supercritical Mach Numbers
Chapter II. Aerodynamic Characteristics of the Wing and Aircraft.
The Effect of Air Compressibility ....
§1. The Dependence of the Coefficient c on the Angle of Attack
§2. The Effect of the Mach Number on the Behavior of the Dependence
c = f(a) ..........
§3. The Permissible Coefficient c per and its Dependence on the
Mach Number . . . . .
%k. Dependence of the Coefficient c on the Mach Number for Flight
at a Constant Angle of Attack ......
§5. The Affect of the Mach Number of the Coefficient c
§6. Wing Wave Drag .........
§7. Interference. .........
§8. The Aircraft Polar. The Effect of the Landing Gear and Wing
Mechanization on the Polar. ......
§9. The Affect of the Mach Number on the Aircraft Polar .
Chapter III. Some Features of Wing Construction .
§1. Means of Increasing the Critical Mach Number
§2. Features of Flow Around Swept Wings
§3. Wing Construction in Turbojet Passenger Aircraft.
§4. Drag Propagation Between Separate Parts of Aircraft
Cfiapter IV. Characteristics of tiie Power System .
§1. Two-Circuit and Turbofan Engines .
§2. Basic Characteristics of Turbojet Engines
§3. Throttle Characteristics ....
§4. High-Speed Characteristics ....
§5. High-Altitude Characteristics
§6. The Effect of Air Temperature on Turbojet Engine Thru
§7. Thrust Horsepower ......
§8. Positioning the Engines on the Aircraft
Chapter V. Takeoff. ......
§1 . Taxi ing .
§2. Stages of Takoff ......
§3. Forces Acting on the Aircraft During the Takeoff
§4. Length of Takeoff Run. Lift-off Speed.
§5. Methods of Takeoff. .....
§6. Failure of Engine During Takeoff .
§7. Influence of Various Factors on Takoff Run Length
§8. Methods of Improving Takeoff Characteristics
Chapter Vl. Climbing ......
§1. Forces Acting on Aircraft ....
§2. Determination of Most Suitable Climbing Speed
§3. Velocity Regime of Climb
%k. Noise Reduction Methods. ....
§5. Climbing with One Motor Not Operating .
Chapter VII. Horizontal Flight
§1. Diagram of Forces Acting on Aircraft
§2. Required Thrust for Horizontal Flight .
§3. Two Horizontal Flight Regimes
%k. Influence of External Air Temperature on
§5. Most Favorable Horizontal Flight Regimes
Altitude and Speed .....
§6, Definition of Required Quantity of Fuel
§7. Flight at the "Ceilings" ....
§8. Permissible Flying Altitudes. Influence of Aire
§9. 'Engine Failure During Horizontal Flight
§10. Minimum Permissible Horizontal Flight Speed.
Requi red Thrust
Forces Acting on Aircraft
Chapter VIM. Descent
§1. General Statements.
During Descent ......
§2. Most Favorable Descent Regimes
§3. Provision of Normal Conditions in Cabin During
High Altitude Flying .....
§^. Emergency Descent
Chapter IX. The Landing .....
§1. Diagrams of Landing Approach
§2. Flight After Entry into Glide Path.
Selection of Gliding Speed ....
§3. Stages in the Landing .....
§4. Length of Post-landing Run and Methods
of Shortening it . .
§5. Length of Landing Run As a Function of Various
Operational Factors .....
§6. Specific Features of Landing Runs on Dry, Ice
Snow Covered Runways .....
§7. Landing with Side Wind ....
§8. The "Minimum" Weather for Landings and Takeoffs
§9. Moving into a Second Circle
Chapter X. Cornering ......
§1. Diagram of Forces Operating During Cornering
§2. Cornering Parameters .....
Chapter XI. Stability and Control abi 1 i ty of Aircraft
§1. General Concepts on Aircraft Equilibrium
§2. Static and Dynamic Stability ....
§3. Controllability of an Aircraft ....
%k. Centering of the Aircraft and Mean Aerodynamic Chord
§5. Aerodynamic Center of Wing and Aircraft.
Neutral Centering .....
§6. Longitudinal Equilibrium ....
§7. Static Longitudinal Overload Stability
§8. Diagrams of Moments .....
§9. Static Longitudinal Velocity Stability
§10. Longitudinal Controllability
§11. Construction of Balancing Curve for Deflection
of Elevator .......
§12. Vertical Gusts. Permissible M Number in
Cruising Fl ight ......
§13. Permissible Overloads During a Vertical Maneuve
§14. Behavior of Aircraft at Large Angles of Attack
§15. Automatic Angle of Attack and Overload Device
§16. Lateral Stability .
§17. Transverse Static Stability
§18. Directional Static Stability
§19. Lateral Dynamic Stability
§20. Yaw Damper
§21. Transverse Controllability
§22. Directional Controllability. Reverse Reaction
§23. Involuntary Banking ("Valezhka")
§2't. Influence of Compressibility of Air on Control
Surface Effectiveness .....•••• 230
§25. Methods of Decreasing Forces on Aircraft Control Levers . . 231
§26. Balancing of the Aircraft During Takeoff and Landing . . 233
Chapter XII. Influence of Icing on Flying Characteristics . . 236
§1. General Statements ......... 236
§2. Types and Forms of ice Deposition. Intensity of
Icing .......••••• 237
§3. Influence of Icing on Stability and Controllability
of Aircraft in Pre-landing Guide Regime ..... 239
Jet-powered passenger aircraft have been adopted and introduced into /Z*
general use in civil aviation.
The first turbojet passenger aircraft built in the Soviet Union was the
Tu-104, and the first foreign turbojets were the De Havilland Comet, the
Sud Aviation Caravelle, the Boeing- 707, the Douglas DC-8, the Convair 880 and
others. These aircraft have been given the name first-generation turbojet
In building the first turbojet passenger aircraft, the designers attempted
to achieve long flight range and to perfect the high-speed properties of the
aircraft, thereby compensating for the heavy fuel consumption required by the
jet engines. The desire to create new aircraft capable of competing with
the old passenger aircraft which were equipped with highly economic piston
engines led to a maximum increase in the lifting capacity, and flight dis-
tance and speed. TTie realization of these qualities became possible only
because of the appearance of jet engines.
Experience in using aircraft has shown that turbojet passenger aircraft
may be economic not only in terms of long-range flight, but for medium- and
even short-range flight as well. As a result, second- generation turbojet
passenger aircraft have appeared: in the Soviet Union there are the Tu-124,
the Tu-134 and the Yak-40, while abroad there are, the De Havilland-121
"Trident", the Bak-1-11, the Boeing- 727, the DC-9'and others. These air-
craft are substantially smaller in dimensions and intended for use on short-
range nets. The high power and low unit load on the wing permit flights
from airfields having relatively short take-off and landing runways.
Turbojet engines surpass piston engines in reliability. With their
short time in series production and use, service periods of 2,000 - 3,000
hours between maintenance checks have been established. This is an important
fact in increasing the economy of using turbojet aircraft, because the cost
of these engines substantially exceeds that of piston engines. In the Five
Year Plan for the development of the Russian economy from 1966 to 1970, the
further development of civil aviation is anticipated and the volume of air /4
travel should increase by a factor of 1.8. New passenger aircraft are going
into service in the airlines.
Turbojet passenger aircraft have flight characteristics which differ from
those of aircraft with piston and turboprop engines in several respects.
These flight features result from the unique high-speed and high-altitude
characteristics of the engines, as well as the flight conditions at these
high speeds and altitudes.
* Numbers in the margin indicate pagination in the foreign text.
With the appearance of jet aviation, there has been a resultant increase
in the importance o£ high-velocity aerodynamics, i.e., the motion of bodies
in air viewed in terms of the effect of its compressibility, i.e., the
properties to change density with a change in pressure. . Ihe first to indicate
the necessity of estimating the effect of air compressibility was the Russian
scientist S.A. Chaplygin, in his work "On Gas Flows" published in 1902. It
was he who developed a method for the theoretical solution of problems of the
motion of gas with allowance made for its compressibility.
The Soviet scientists Academicians S.A. Khristianovich, M.V. Keldysh,
A. A. Dorodnitsyn, Professors V.S. Pyshnov, F.I. Frankl', I.V. Ostoslavskiy,
B.T. Goroshchenko, Ya.M, Serebriyskiy, A. P. Mel'nikov and others, through
their studies in the field of high- velocity aerodynamics, contributed much
which was of great value in the design of high-speed aircraft.
The Soviet turbojet passenger aircraft created by aeronautical engineers
A.N. Tupolev, S.V. Ilushin and A.S. Yakovlev, take their places in the ranks
of the first-class aircraft.
The successful use of new aviation technology by flight and engineering
personnel is unthinkable without a deep understanding of the laws of aero-
dynami cs .
Aircraft aerodynamics, when thought of in terms of the flight crew, is
usually called practical aerodynamics. The ntmiber of problems involved in
aerodynamics is quite substantial. These include studying the laws of the
motion of air and the interaction o£ air flows with bodies moving in them,
the interaction of shock waves with various parts of the aircraft, aircraft
flight dynamics as affected by the forces applied to the aircraft (including
aerodynamic forces), and aircraft stability and handiness.
It is the object of this book to examine these questions in tenns of
turbojet passenger aircraft.
NASA TT F-5A2
THE PHYSICAL BASIS OF HIGH-SPEED AERODYNAMICS
ABSTRACT. This book presents the physical bases of high-
speed aerodynamics, and the influence of air compressibility
on the aerodynamic characteristics of wings and aircraft.
Primary attention is turned to passenger jets. The following
areas are covered; takeoff characteristics of jets and
methods of Improving them; best climbing modes; horizontal
flight; the descent; the landing approach; turns and corners;
controllability and stability; icing and its influence on
flying characteristics; and the characteristics of modern
§ 1. Variations in the Parameters of Air with Altitude. The Standard
The flight of aircraft, like that of other flight vehicles, is affected /5
by the condition of the atmosphere -- the shell of air surrounding the earth.
TTierefore, it is quite vital to know the processes occurring in the atmos-
Only the atmosphere's lower boundary, the earth's surface itself, is
clearly delineated. The upper atmosphere is more difficult to establish
because the density of air decreases constantly with altitude and even at an
altitude of ,100 km it measures approximately one millionth that on the earth's
surface. Normally, the upper limit of the atmosphere is considered the
altitude at which the air density approaches that o£ the gases filling inter-
Data fran direct and indirect observations show that the atmosphere has
a layered structure. In 1951 the International Geodesic and Geophysical Union
adopted the division of the atmosphere into five basic spheres or layers:
the troposphere, the stratosphere, the mesosphere, the thermosphere and the
The Troposphere is the lowest layer of the atmosphere, which in the middle
latitudes extends to an altitude of 10-12 km, in the tropics — to an altitude
of 16-18 km, and in the polar regions -- to an altitude of 8-10 km. This
layer is of tremendous practical interest in aviation, because all the most
important phenomena encountered by the pilot occur basically in the tropo-
sphere. It is here that the formation of clouds and fogs, the fall of
precipitation, and the development of storms occur.
The most significant feature of the troposphere is the decrease in
temperature with a rise in altitude (averaging 6.5° per km of altitude). The
troposphere is the area of thermal turbulence resulting from the unequal
heating of layers of air at the earth's surface and at various altitudes, as
well as the dynamic turbulence resulting from the friction of the air with
the earth's surface and its intense vertical displacement at the boundaries /5_
between cold and warm air masses of atmospheric fronts.
The troposphere ends in the layer of the tropopause. The thickness of
the tropopause fluctuates from a few hundred meters to several kilometers.
It is usually a continuous layer which surrounds the earth's sphere itself,
while its altitude and temperature are functions of the geographic latitude,
the time of year and the atmospheric processes developing. Over the equator
and its neighboring areas, the tropopause is located at an average altitude
of 16-18 km (India), while in the middle latitudes it is located at an
altitude of 10-12 km, and in the polar regions it has an altitude of 8-10 km,
while over the pole it may drop to 5-6 km. Jet aircraft normally fly close
to the limit of the tropopause, a characteristic feature of which is the
existence of cyclic bumps beneath the tropopause itself.
The stratosphere is located above the tropopause and extends to approxi-
mately an altitude of 35-40 km. Constant temperature with altitude is
characteristic of its lower layers. The insignificant content of water vapor
in the stratosphere results in the lack of clouds from which precipitation
would fall. According to data from pilots who have flown at altitudes of
12-16 km, in the lower stratosphere it is most frequently cloudless. The air
is stable and vertical motion is slight. This aids in smooth flight. There
is seldom bumpiness , and only then close to the tropopause.
The mesosphere runs from the upper boundary of the stratosphere to an
altitude of 80 km.
The thermosphere is located above the mesosphere and extends to an
altitude of 800 km.
The exosphere is the outer layer of the atmosphere, or the dissipative
layer, and is located above the thermosphere. Gases here are so rarefied and
at the high temperatures observed there have such high velocities that their
particles (heliian and hydrogen) break away from the earth's attractive force
and move into interplanetary space.
Thus we have a brief description of a structure of the atmosphere.
Atmospheric conditions are characterized by the various meteorological
elements -- atmosphere pressure, temperature, humidity, cloud cover, precipi-
tation, wind, etc. The atmosphere may be characterized as a variable medium.
As a result of unequal heating of the air masses at the equator and poles,
flows are formed which result in the passage of cold air toward the equator and
warmer air toward the poles. The effect of the earth's rotation in the
northern hemisphere causes the air flow to deviate to the right and move from
the south to the southwest, while approaching 30° N it moves to the west.
Therefore, flights from west to east over the territory of the USSR are /7
accompanied by tail winds, while east- to-west flights encounter head winds.
The shift from westerly winds to easterly occurs at altitudes around 20 km.
Whereas piston aircraft fly only in the lower troposphere, jet aircraft, in
contrast, fly in the upper and -- to a certain extent -- in the lower strato-
The further development of high-speed aviation will in the near future
permit us to fly at supersonic speeds corresponding to Mach = 2.5-3. At this
point, flights will be in the stratosphere.
Before the perfectioning of jet aircraft, it was assumed that at high
altitudes the flights would encounter favorable weather conditions. However,
it was found that at altitudes of 10,000 - 12,000 m cloud cover and bumpiness
were sometimes encountered. To these well-known phenomena, there were added
the jet streams characteristic of altitudes of 9-12 km.
The jet streams are the broad expanses of zones of very strong winds
observed in the upper layers of the troposphere, usually at altitudes of
9000 - 12,000 m. Post-war studies showed that the minimum velocity of the jet
stream (along its axis) equalled approximately 100 km/hr, while the maximum
was 750 km/hr (over the Pacific Ocean). Over the USSR, the wind speed in the
jet stream reaches 100 - 200 and sometimes even 350 km/hr, while over the
North Atlantic and Northern Europe it reaches 300 - 400, 500 over the USA,
and 650 km/hr over Japan. The jet stream is comparable to a gigantic highly
oblate channel with a height averaging 2-4 km and a width of 500 - 1000 km.
These flows run basically west-east, but in certain sections they may vary
Flight speed may be increased by the selective use of jet stream tail
winds, while flight against the head wind should be one or two km above or
below the axis of this stream. As a rule, the jet streams are to be found in
the region where the tropopause is situated.
In studying aircraft flight and determining the forces acting on aircraft,
we may consider the air as a continuous medium.
At sea level, the air consists of a mixture of nitrogen (78.08% of the
volume of dry air), oxygen (20.95%) and insignificant quantities of other
gases (argon, carbon dioxide, hydrogen, neon, helium, etc.). The air also
contains water vapors .
In the troposphere and stratosphere the temperature, pressure and
density of the air vary within rather broad limits as a function of the geo-
graphic latitude of the locale, the time of year, the time of day and the
In order to achieve a common concept of the characteristics of the
atmosphere (pressure, temperature and density), the standard atmosphere was
arrived at -- the arbitrary distribution, in the atmosphere, o£ pressure,
density and temperature for dry, clean air (containing neither moisture nor
dust) of a constant composition applicable for engineering — primarily
aviation — calculations with respect to their comparability (for example, in
calculating the lift and drag and for graduating various aerial navigation
instruments such as altimeters and others).
In the standard atmosphere, the altitude is computed from sea level.
Normal conditions at sea level are: atmospheric pressure p. = 760 mm Hg, air
density p = 0.125 kG • sec /m , temperature t„ = 15 C (or T„ = 288°K) and
specific weight of the air -y = 1.225 kG/m .
Variations in air pressure and density with altitude, which proceed in
accordance with a specific law, are calculated per each altitude according to
special formulas. The air temperature in the standard atmosphere up to an
altitude of 11,000 m drops uniformly by 6.5°C per 1000 m. Above 11,000 m,
the temperature is considered constant and equal to -56.5°C. In fact, how-
ever, at this altitude it may reach -80°C. Results of calculations are
given in the table. Below we present an abbreviated table of the standard
TABLE 1. STANDARD AIMOSPHERE (SA)
Relative! „ .
i (V ' °c
A = %
Tr. Note: Commas indicate decimal points.
§ 2. Compressibility of Air /9
Compressibility is the property of gases (and fluids) to change their
initial volume (and, consequently, density) under the effect of pressure or a
change in temperature.
In solving technical problems, compressibility is taken into account in
those cases when changes in volume (density) are considerable by comparison
to the initial volume (density) ,
If the volume of water with an increase in pressure of 1 at. with
constant temperature changes an average of only 1/21,000 of its initial value,
i.e., only 1/210 of a percent, air, which has a high compressibility, requires
a change in pressure of only one one hundredth that of atmosphere (0.01 at.)
to change its volume by 1% under normal atmospheric conditions.
Therefore, all gases are considerably more compressible than dropping
liquid. For example, if the pressure in a given mass of gas increases in
such a way that its temperature does not vary during this change, the volume
of the gas decreases. When the initial pressure is doubled, the volume
decreases by 50%. The change in volume for gas is equally high during heating.
Differences in compressibility of liquids and gases are explained by
their molecular structure. In liquids, the inter-molecular distance is small,
i.e., the molecules are rather dense, which determines the small capability
liquids have of compressing. By comparison with liquids, gases have an
extremely low density. For example, the density of water is 816 times that of
air. The low density of air and other gases is explained by the fact that in
gases the inter-molecular distance substantially exceeds the dimensions of
the molecules themselves. Therefore, when there is an increase in the pressure,
the voliome of the gas decreases due to the decreasing distance between
molecules. Thus arises the elasticity which gas possesses.
In aviation problems, the need to account for air compressibility results
from the fact that at high flight speeds in air, substantial differences in
pressure arise which are the cause of substantial changes in its density.
To evaluate the effect of compressibility, let us examine the speed of
§ 3. The Propagation of Small Disturbances in Air. Sound and Sound Waves.
The property of compressibility is intimately related to the phenomenon
of the propagation of sound in gases. The speed of the propagation of sound
plays a vital role in high-speed aerodynamics. The effect of compressibility
on the aerodynamic characteristics of aircraft is a function of the degree
to which the flight speed of the aircraft approaches the speed of soiond. When
air flows at speeds greater than the speed of sound, qualitative changes occur /lO
in the character of the flow.
The sensation which we perceive as sound is the result of the effect, on
our auditory apparatus, o£ the oscillatory motion of air caused, for example,
by the motion of some body in it. The displacement of each particle of air
during its vibration is insignificantly small. The particles vibrate around
their equilibrium configuration, which corresponds to their initial state.
However, the laboratory process is propagated a very long distance.
The human ear perceives as sound those disturbances which are transmitted
with a frequency from 20 to 20,000 vibrations per second. Those with a
frequency of less than 20 per second are called infrasound, and those above
20,000 per second are called ultrasound.
By small disturbances we mean slight changes in the pressure and density
of the medium (gas or liquid) . Disturbances being propagated in the medium,
such as air, are called waves (due to the similarity of this phenomenon to
waves on the surface of water) .
The speed of the propagation of the disturbances in space (the wave
velocity) is quite substantial. The speed of propagation of a sound wave,
i.e., small changes in density and pressure, is called the speed of sound.
It is a function of the medium in which the sound is being propagated and
of its temperature.
In high-speed aerodynamics, sound is considered as waves of perturbations
created in the air by a flying aircraft.
The speed of sound in gases is a function of temperature. The higher the
gas temperature, the less compressed it is. Heated gas has a high elasticity
and therefore is more difficult to compress. Cold air is easily compressed.
For example, at a gas temperature T = (or t = -273°C), the speed of sound
equals zero because under these conditions the gas particles are immobile and
exercise only slight disturbances, with the result that they can create no
The dependence of the speed of sound in air on temperature may be
determined according to the following approximate formula:
a = 20 /r m/sec.
Within the limits of troposphere, the air temperature decreases with
altitude. Consequently, in the troposphere the speed of sound also decreases
with altitude. On the earth's surface under standard conditions (p = 760 mm
Hg, t = 15 sec), a = 340 m/sec. With an increase in altitude for every 250 m, /ll
the speed of sound decreases by 1 m/sec.
At altitudes above 11,000 m, the temperature is (according to the
standard atmosphere) considered constant and equal to -56.5°C. Consequently,
the speed of sou nd at these altitudes should also be considered constant and
equal to a = 20 v/273 - 56.5 = 296 m/sec (Fig. 1).
§ k. The Speed of Sound as a Criterion for the
Compressibility of Gases
In gas dynamics, for the speed o£ sound
there is the well-known formula:
290^300 310 3Z0 J50(j,m/gec.
Figure 1. The Change in
th-= Speed of Sound with
where Ap is the change in pressure, Ap
change in gas density which it causes,
compressed the gas is, the slower the speed of
sound, so that one and the same change in density
may be obtained through a slight change in
pressure. And, in contrast, the less the com-
pressibility of the medium and the greater its
elasticity, the greater the speed of sound in
the same medium. In this case, a slight change
in density may be achieved only through a great
change in pressure. The speed of sound is taken
into consideration in any case in which there is an evaluation of the effect of
compressibility in any aerodynamic phenomena, because the value of the speed of
sound characterizes the compressibility of the medium. If the medium is
elastic (compressible), compressions and expansions will vary substantially
from layer to layer with the speed of sound. If the medium is absolutely
incompressible, i.e., for any increase in pressure the volume or density
remains unchanged, then as can be seen from the formula given above, the speed
of sound will be quite high. In such a medium, any disturbances are propa-
gated any distance instantaneously.
As was shown above, the value of the speed of sound varies in different
gases and, in addition, it is a function of temperature. With an increase in
altitude, temperature and the speed of sound decrease. Therefore, the effect
of compressibility on the flight of aircraft at high altitudes should appear
even greater. Let us introduce several values for the speed of sound at
t = 0°C: for nitrogen it is 337.3, for hydrogen it is 1300, and for water it
is 1450 m/sec.
For solid bodies, which are less compressible than gases, the speed of
sound is still greater. Thus, in wood the speed of sound is 2800 m/sec, while
in steel it is 5000 and in glass it is 5600.
An aircraft in flight, repelling air on all sides, partially compresses
it as well. At low flight speeds, the air in front of the aircraft succeeds
in being displaced and adapts itself to the flow around the aircraft so that
compression is insignificant in this case. At higher flight speeds, however,
the air compression begins to play a more important role. In this case, there-
fore, for a scale of flight speed we must use a characteristic speed which may /12
serve as a criterion for the compressibility of the medium. Such a speed is
the speed of sound, inasmuch as it is a function of the temperature and
properties of the gas .
§ 5. The Mach Number and its Value in Flight Problems
The ratio of the flight (or flow) speed to the speed of sound is called
the Mach niimber:
M = X.
Let us assume that the true flight speed (see § 6 of this Chapter) of an
aircraft at an altitude of 10,000 m is 920 km/hr (255 m/sec) . Then the Mach
number M = -s^r^r = 0.85, where a = 300 m/sec. In other words, the flight speed
is 85% of the speed of sound at this given altitude.
Thus, in comparing the speed of the motion of the body in the air with
the speed of sound under the same conditions, we may determine the effect of
air compressibility on the character of the flow around the body. The Mach
number is the index of the air compressibility. The greater the Mach nimiber,
the greater the air compressibility should be during flight.
To monitor the Mach number in flight, an instrument -- the Mach indicator
(Machmeter) -- is usually set up on the pilot's instrument panel. In high-
speed flight, especially when maneuvers are being performed which result in
a loss of altitude, the reading on this instrument must be followed, and the
pilot must not exceed the Mach number which the instructions permit for the
given aircraft. If flight speed remains constant as altitude increases, the
Mach number will increase due to the decrease in the speed of sound.
Failure to monitor the Mach nimiber in jet aircraft would result in grave
trouble because knowing the indicated speed (see § 6 of this Chapter) and even
the true speed does not give the pilot a full understanding of the flight Mach
number at any specific altitude. For example, if the aircraft is flying at an
indicated speed of 500 km/hr at an altitude of 12,000 m, the true speed will
be around 930 km/hr while the speed of sound is 1063 km/hr, so that under
these given flight conditions the Mach number = 0.875. If, however, the
aircraft is flying with an indicated speed of 500 km/hr at an altitude of
1000 m, the true speed is only 525 km/hr, while the Mach number = 0.43.
In turbojet aircraft, a change in the Mach number may be represented in
the following way. After takeoff and retraction of the landing gear and
wing flaps, the aircraft picks up speed until it achieves an indicated speed
of 500 - 600 km/hr and starts climbing. Starting at an altitude of around
1000 m, the Machmeter shows a Mach number of M= 0.5 -0.55. As the aircraft
climbs, the true speed will increase, the speed of sound will decrease, and /15
the Mach number increase. When the aircraft reaches an altitude of 8-9 km,
the Mach number reaches a value of 0.63 - 0.66 (depending on the actual
temperature at that altitude). At altitudes of 10-12 km, during acceleration
the Mach number increases to 0.80 - 0.85. At high altitudes the Mach number
will be greater when the same true speeds are maintained. Turbojet aircraft,
like many other high-speed aircraft, have a limit to their Mach number becaxise
of conditions of stability and handiness (more will be said concerning the
selection of the Mach number in Chapters 7 and 11). Therefore (especially at
high altitudes) , it is insufficient to monitor flight simply with respect to
speed; the Mach indicator mvist also be observed.
§ 6. Flight Speed. Corrections to Instrument Readings Necessitated by
Compress! bi 1 i ty
Aircraft speed indicators measure directly not only the speeds, but the
velocity head q = pV /2. The actual flight speed is not the same as this
speed, which is indicated by the instrument, because the air-pressure sensor
indicates the effect of perturbations created by the aircraft and the air
compressibility. In addition, the value of the actual flight speed depends
on instrumental corrections.
Therefore, to eliminate the above-mentioned errors in the instriiment
readings, the following corrections are introduced: aerodynamic, which
accounts for the difference in the local pressures (at the point where the
air-pressure sensor is located) from pressures in the undisturbed incident
flow, corrections for compressibility, and instrument corrections*.
The speed which would be shown on an ideal (i.e., error-free) speed
indicator is called the indicated speed V.. The speed which is read from the
instrument (read from the wide needle) , does not as a rule equal the indicated
speed. Therefore, a special name has been created for it -- instrument speed
The true air speed is the speed of the aircraft's motion relative to the
air (and is read from the thin arrow on the instrument) .
The KUS-1200 combined speed indicator, which jet aircraft flying at
Mach speeds up to 0.9 are equipped with, shows the instrument speed and the
true air speed. During low-altitude flight (where the air density is close
to that of the earth's surface, equal to 0.125 kG • sec2/m4) , the instrument
and true air speeds agree and both arrows on the instrument move together,
being superimposed. With an increase in altitude, the true air speed
surpasses the instrument speed and the arrows diverge, forming a "fork." /14
Knowing the true air speed and wind speed, it is possible to determine the
ground speed, i.e., the speed of the aircraft's displacement relative to the
earth. In flying and aerodynamic computations, both the indicated and
instrument speeds are used. And what is the difference between them? To
switch from instrument speed to indicated speed, we must introduce an aero-
dynamic correction and a correction for air compressibility:
* M.G. Kotik, et al.. Flight Testing of Aircraft, Mashinostroyeniye, 1965
(Available in NASA translation).
V. ^ = V. + 6V + 6V = V. + 6V ,
mst 1 a comp i a
6V = correction for compressibility, and
V. = indicated groimd speed.
For high-speed aircraft, an essential correction is the correction for
air compressibility, whose value may range from 10 to 100 km/hr. The effect
of air compressibility increases the speed indicator reading, so that (SV
is always negative (Fig
MB v^ , km/hr
Nomogram for Determining the Correction for
Ai r Compress ibi 1 i ty
The aerodynamic correction may reach values from 5 to 25 km/hr and may be /15
either positive or negative. Whereas the correction for compressibility is
identical for all aircraft, the aerodynamic correction is basically a function
of the type of aircraft or, more specifically, the position and features of
the engine. Therefore, each aircraft has its own graph of aerodynamic
The indicated speed with the correction for compressibility is called the
indicated ground speed: V. = V. + 6V . At sea level, irrespective of air
" '^ 1 1 comp ^
temperature, V. = V. . According to the nomogram in Figure 3, we may find the
flight Mach number being given the value of V. , and then determine the true
flight speed: V = aM. For example, we must determine the true speed and
flight Mach number for the aircraft if at an altitude of 10,000 m, V. =
= 500 km/hr. Taking the aerodynamic correction 6V = -10 km/hr, we find:
V. = 490 km/hr. For this speed, according to the nomogram (Figure 2) , we
obtain iSV = -23 km/hr. Then let us determine the indicated speed V. =
comp ^ 1
V. - 10 -23 = 500 -33 = 467 km/hr. The true flight speed may be found from
the following expression:
where for H = 10,000 m, A = 0.337, a \/K = 0.58 (see the table for the /16
standard atmosphere). Or, for speed V. = 490 km/hr, according to the nomo-
gram (Fig. 3), we obtain a Mach number of 0.75. Knowing the speed of sound at
H = 10,000 m and the flight Mach number, it is easy to- determine the true
speed: V^ = aM = 300 • 0. 75 • 3.6 = 810 km/hr.
The accepted value 6V = -10 km/hr is characteristic of modem high-
speed aircraft within the range of their indicated speeds of 220 - 600 km/hr.
Later we will determine the correction for air compressibility in each
concrete case according to the nomogram in Figure 2, while we will assume
that the aerodynamic correction is 6V = -10 km/hr.
§ 7- The Character of the Propagation of Minor Perturbations in Flight
at Various Altitudes
In an example of aircraft flight, let us examine the manner in which
slight fluctuations in density and pressure, i.e., minor perturbations, will
be propagated in the air flow. The aircraft, being the source of the per-
turbations, has an effect on the air particles located in front of it and
perturbations are sent forward from one particle to the next at the speed of
Let us first take an aircraft flying at below the speed of sound (Fig. 4a).
m vjo 500 600 700 800 300 MB im mo ma mo mo y km/hr
Figure 3. Nomogram for Determining the Mach Number
N Spheric .^ ^— — ^
\ sound !/ , ^
I wave B
'^f\- 4f — u^ — r^'-'
--i^^ \ /-? propaggcion
unperturbed VsT"^-^'^ line/''
Figure 4. Propagation Characteristics for Sound Waves
When the aircraft passes through point A the perturbations created by it
at that given moment, propagating along a sphere at the speed of sound, over
take the aircraft. After a short time, the Mach wave reaches point B, while
during this time the aircraft has succeeded only in progressing to point C;
thus, its flight speed is below the speed of sound. Passing through point D,
it again creates perturbations which will be propagated with the speed of
sound and in a short while reach point E. The aircraft, however, during this
time will not have reached point E but will be located between points C and
E. Thus, the aircraft remains constantly within the sphere created by its
sound wave. If, however, the aircraft flies at the speed of sound (Fig. 4b),
then point B is reached simultaneously by both the aircraft and the sound
waves, i.e., the perturbations created by it at points A, C and D.
Thus, in front of the aircraft there are always Mach waves which,
becoming superimposed upon each other, form a dense section of air called the
compression shock or shock wave.
If the aircraft flies above the speed of sound, it moves ahead of the
spherical waves it has created (Fig. 4c). The aircraft will reach point C
at the moment when the perturbation it created at point A has reached only
point B, while the perturbation created at point D has reached point E. Thus,
behind an aircraft flying at supersonic speed a Mach cone is formed which
consists of an infinite number of Mach waves propagated along the sphere at
the speed of sound. However, the air mass within the Mach cone is displaced /17
relative to the earth at the aircraft's speed. The greater the aircraft's
speed, the sharper the angle at the tip of the Mach cone. This angle is
determined according to the formula (Fig. 4c) :
sin ^ = -^ .
If the Mach number is 1, then 4) = 90°, while the full angle is 180° (normal
shock); for M = 2, sin cj) = 0.5 and the angle <i> = 30° (full angle 60°).
Compression shocks are both normal and oblique. A normal compression
shock is one whose surface is perpendicular to the direction of the incident
flow, i.e., which forms an angle 3 = 90° with it (Fig. 5a). Oblique shocks
are those whose surface forms an acute angle of g < 90° v<rith the direction
of the incident flow (Fig. 5b).
The greatest speed losses and increases in pressure are observed when
the flow passes through a normal compression shock. The braking of the flow
on this shock is so substantial that behind the shock the flow velocity must /18
be below the speed of sound (by as much as it was above the speed of sound
in front of the shock) .
In an oblique shock the losses are less than with a normal shock,
specifically, proportionately little the more the shock was inclined in the
direction of the flow, i.e., the less the angle 3. The intensity of an
oblique shock is also substantially less than a normal shock. If the angle B
is close to 90°, then behind the oblique shock the speed of the flow is
subsonic, while somewhat greater than that which would be obtained if the
shock were normal.
Figure 5- Formation of Normal (a) and Oblique
(b) Compression Shocks.
through an oblique shock
change the direction of
their motion, deviating,
from their initial
direction. During flow
around a wing or fuselage
with a speed exceeding the
speed of sound, an oblique
shock developes in front
of the wing or fuselage.
for trans- and super-
sonic speeds must have
aerodynamic shapes which
do not generate normal
compression shocks. The
forward edge of the wing
on supersonic aircraft
must be knife-like, and
the wing itself must be
§ 8. Trans- or Supersonic Flow of Air Around Bodies
In the case of low- velocity flow around bodies, the flow is deformed at
a substantial distance from the body and air particles, in breaking away, flow /19
smoothly around it (Fig. 6a)
Subson i c
a Wi ng Prof i le.
When this occurs, the pressure close to the
body varies insignificantly, which permits us
to consider air density as constant. As a
result of the difference in pressures under
and over the Wing, left is created.
In the case of sonic or supersonic flow
around a body, local air pressure and density
variations arise which, propagating at the
speed of sound, form a sonic or supersonic
shock wave in front of the body.
This occurs because the speed of the air
particles close to the body suddenly varies
in both amount and direction. When this
occurs, the flow in a sense "encounters" an
obstacle which, depending on the situation,
may be the body itself or an "air cushion" in
front of it and form a compression shock
(shock wave) . At this compression shock there is an uneven change in the
basic parameters characterizing the conditions o£ the air, i.e., speed V,
pressure p, density p and temperature T. Shock waves may be formed either
in front of the profile or close to its trailing portion. Precise calcula-
tions and measurements have shown that the thickness of the shock waves or
compression shocks is negligibly small and has an order of length of the free
path of the molecules, i.e., 10-4 _ iq-S mm (0.0001 - 0.00001 mm).
§ 9. Sonic "boom"
Supersonic flight is accompanied by the characteristic sonic "boom."
This phenomenon is the result of the formation of a system of compression
shocks and expansion waves in front of the nose of a fuselage, the cabin, or
where the wing and tail assembly join the fuselage.* The most powerful shock
waves are formed by the aircraft's nose and wing, which during flight are the
first to encounter the air particles, and the tail assembly. These shock
waves are labeled bow and tail shock waves, respectively (Fig. 7a). Inter-
mediate shock waves either catch up with the bow shock and merge with it or /20
fall behind and merge with the tail shock.
Behind the bow shock, the air pressure increases unevenly, becoming great-
er than atmospheric pressure, and then decreases smoothly and becomes even less
than atmospheric, after which it again increases unevenly until it is
practically atmospheric again at the tail wave.
The sudden pressure drop is transmitted to the air around it in a
direction perpendicular to the wave surface. Persons on the ground feel this
drop as a strong "boom." Sometimes a second 'Tjoom" is heard -- this is the
result of the successive effects of both the bow and tail shock waves.
Figure 7. Air Pressure Changes during a "boom" in
tlie Vertical Plane below the Aircraft (a), and the
Interception of the Conic Shock Wave with the Earth's
A.D. Mironov, Supersonic "Floe" in Aircraft. Voyenizdat, 1964.
Repeated observations have established that the two successive sonic
booms are distinctly heard only when there is more than l/8th o£ a second
The longer the aircraft, the longer the time interval between the
occurrence of the bow wave and the tail wave. Therefore, two "booms" are
distinctly heard in the case of an aircraft with a long fuselage. And, in
contrast, an only vaguely separated "boom" indicates that the aircraft has
small dimensions or is flying at a relatively low altitude.
If the aircraft flies at a constant supersonic speed, the 'Tjoom" is
heard simultaneously at different points on the earth's surface. If these
points were to be joined by a line, we would obtain a hyperbola forming as
a result of the interception of the conic shock wave with the plane of the
earth's surface (Fig. 7b). One hyperbola corresponds to the bow wave, and
the other -- to the tail wave. The lines of simultaneous audibility of the
"boom" are displaced along the earth's surface, following behind the air-
craft and forming unusual trails. At the same time, directly below the air-
craft there is a substantially louder "boom," which attenuates as a function /21
of distance and under certain circumstances it is completely inaudible. The
ground observer who hears the "boom" from an aircraft flying, let us say, at
an altitude of 15 km with a speed twice that of sound will not observe the
aircraft above him; at an altitude of 15 km, it takes sound approximately
50 sec to reach the ground at an average speed of 320 m/sec, while during
this time the aircraft will have covered approximately 30 km.
To get an idea of the effect of a pressure drop on building structures,
let us point out that the overpressure Ap = 10 kG/m^ creates a short- lift
load of 20 kG on a door with an area of 2 m2, for example. A fighter with a
fuselage length of 15 m at Mach 1.5 and H = 6000 m creates Ap = 11 kG/m^. A
heavy, delta-winged supersonic aircraft weighing 70 tons will, flying at an
altitude of 20 km and at Mach 2 create Ap = 5 kG/m^, and at low altitudes
(5-8 km) a drop may reach 12-18 kG/m2. it is a known fact that in their
design, buildings are planned for the so-called wind load, which corresponds
to the force of the pressure of air moving at a speed of 40 m/sec, i.e.,
greater than 140 km/hr. This type wind will create an overpressure of 100 kg
on 1 m^ of wall surface. The pressure in the "boom" at permissible flight
altitudes is l/5th or l/6th that of the design allowance for wind load.
The characteristics of the effect of pressure drops in shock waves during
"booms" are given in Table 2. For example, on a wall with an area of 12 m^
during an overpressure of 50-150 kG/m^, there is a short-lived load of 600-
1800 kG. Under the effect of such a load, wooden structures may collapse.
Therefore, aircraft are forbidden to accelerate to supersonic velocities below
9-10 km over populated areas. In the opinion of foreign specialists, a sonic
"boom" with an intensity of 5 kG/m2 is the most which can be tolerated
harmlessly. Therefore, future supersonic jet aircraft with heavy flight
weights (140 - 170 tons) will have to fly at altitudes of 18-24 km in order
to minimize the effect of pressure drops. In this case, they will have to
climb to altitudes of 9-10 km at subsonic light regimes (Mach niomber = 0.9 - /22
0.92), while beyond that at up to scheduled flight altitude at Mach M = 1.0 -
1.2, and only at this altitude will they be able to accelerate to supersonic
Relative Loudness and Resultant Destruction
0.5 - 1.5
1.5 - 5
Close blast or thunder
5 - 15
Very close, loud thunder (window glass rattles
15 - 50
Large window panes shatter
50 - 150
Light structures collapse
The sound o£ the sonic boom is a function of the flight altitude, Mach
niimber, aircraft's angle of attack, flight trajectory, atmospheric pressure
at sea level and at the flight altitude, and wind direction with respect to
altitude. For example, the "boom" from an aircraft flying at an altitude of
15 km and at Mach 2 (V = 2120 km/hr) is heard to a distance of 40 km from the
aircraft's path, while at an altitude of 11 km it is heard only to a distance
of 33 km. During flight at an altitude of 1.5 km at Mach 1.25, the "boom"
is heard only within a belt 8 km wide.
A tail wind may displace the shock wave, resulting in displace of the
audibility zone. The climbing and descent speeds and the angle of inclination
G of the trajectory have significant effects on the size of the audibility
zone and the loudness of the "boom." For example, in gaining altitude at an
angle of 6 = 15° at H = 5 km, the "boom" is heard on the ground at M > 1.2.
In descending from an altitude of 10-11 km at an angle Q = -10°, the "boom"
reaches the .ground only at M = 1.03.
In conclusion, let us dwell on the effect of the shock wave created by
a supersonic aircraft on a passenger aircraft in flight. As has already been
said, the pressure drop during a compression shock is 5-18 kG/m2. If for the
mean value we select 10 kG/m2, it amounts to less than 0.1% of the air
pressure at ground level (p = 10,332 kG/m2 = 1 at.). The velocity head for
a jet passenger aircraft flying at a speed of 850 km/hr and at an altitude
of 10 km is approximately 1200kG7m , i. e. , more than 100 times the pressure
drop in the "boom." Consequently, such a drop has essentially no effect on
an aircraft in flight. However, there may be a certain effect on the air-
craft's behavior as created by the accompanying jet from the aircraft flying
by; this effect is comparable to that of a slight gust (a single gust of
"bumpy air") , directed along the propagating line of the shock wave. As a
result, the aircraft will experience slight bumpiness.
§ 10. Features of the Formation of Compression Shock during Flow Around
Various Shapes of Bodies
Let us now look at the features of the foimation of compression shocks
first with the example of flow around the air inlet of a jet engine during
supersonic flight, and then let us consider flow around the profile.
The existence of a noimal shock at the intake to the diffuser leads to
substantial losses of total pressure (kinetic energy) of the air entering
the compressor and the combustion chamber.
During deceleration in the diffuser, the supersonic flow is transformed
as it passes through the normal compression shock. When this occurs, one
part of the kinetic energy of the air is used for its compression, while the /25
other is transformed into heat (lost energy) . However, during flight of
the Mach number M < 1.5, losses at the shock are small. As a rule, therefore,
for such flight speeds intake devices are used on subsonic aircraft.
At flight greater than 1.5 Mach, however, losses at the normal shock
beccme greater. To eliminate this, the process of air deceleration in the
intake device is achieved through the creation of systems of oblique shocks
which terminate in a weak normal shock. Because overall energy losses in
a system of oblique shocks are less than in one normal shock, the pressure at
the end of the deceleration will retain a high value. Thus, the normal shock
is divided into a series of oblique shocks. Structurally, this is achieved
through setting up in the diffuser a special spike in the shape of several
cones whose tips are directed according to flight (Fig. 8a).
When flight speed is decreased, the angles of inclination of the oblique
shocks increase (the angle B tends toward 90°; see Figure 5). As speed is
increased, the reverse occurs, and these angles decrease. This hinders the
operation of the input device inasmuch as the front for all the shocks will
not pass through the inpvit edge of the cone (Fig. 8b). Therefore, sometimes
the spike is adjustable, so that in the event of changes in speed, its
position can be varied axially, thereby helping the shock to pass through the
leading edge of the air intake at all flight speeds.
On the wing profile, the formation of compression shocks occurs even
substantially below the speed of sound. As soon as the flow speed of the
convergent stream exceeds the speed of sound somewhere on the profile, Mach
waves appear which, in accumulating, form a shock. It must be noted that
this shock wave is formed first on the upper profile surface close to some
point corresponding to the maximum of the local speed and the minimum
pressure on the profile. As soon as the speed of the flow surpasses the speed /24
of sound, a shock wave forms on the lower profile surface as well (Fig. 9).
1. At point C the point of least pressure on the profile, the speed of
the motion of the air has attained the local speed of sound (Fig. 9a). The
Mach waves move from the source of the perturbation toward point C and,
running into each other, form a weak normal compression shock.
Figure 8. Formation of Compression Shocks at the Intake to
the Diffuser of a Turbojet Engine at Supersonic Flight Speeds:
a - line drawing of input device with cone: OA, BA -- oblique
compression shocks, AK -- normal compression shock; b -
operational configuration of supersonic diffuser during flight
speed below its design speed.
^5 f-a ^, shock
Figure 9. The Formation of Compression Shocks at Various
2. As the speed of sound increases somewhat (at V„ > V,), the speed
of the flow around the profile increases (Fig. 9b). Behind point C, the
speed of the flow becomes greater than the speed of sound. A section
appears where the flow moves at supersonj-C velocity, resulting in the
formation of an oblique shock.
3. At a speed o£ V_ (V_ < a) , regions o£ sonic and supersonic flow also
form on the bottan o£ the profile, resulting in the formation of compression
shocks (Fig. 9c).
4. At a speed of V. close to the speed of sound, the compression shocks
are displaced toward the trailing edge, thereby increasing the section of the
profile which encounters supersonic flow past it (Fig. 9d) .
5. When velocity V^ becomes somewhat greater than the speed of sound, a
bow wave forms in front of the profile and a tail wave forms behind it (Fig.
During flow around a blunted body, the compression shock forms at a /25
slight distance from its forward section and assumes a curvilinear form
(Fig. 10a). At its forward edge, the shock is normal -- here it is perpen-
dicular to the incident flow. Depending on the distance from the body, the
angles of inclination of the shock decrease. During supersonic flow aroimd
a knife-edged body such as a wedge with a large open angle (Fig. 10b), the
shock is formed also at a slight distance from the bow point and also has a
curvilinear form. If the open angle of the wedge is small enough, the
compression shock "seats itself" on the sharp edges (Fig. 10c).
Figure 10. The Formation of Compression Shocks at Identical
Flow Velocities: a - in front of a blunted body, b and c -
in front of knife-edged bodies.
§ 11. Critical Mach Number. The Effect of Compressibility on the
Motion of Air Flying Around a Wing
The compressibility of the air begins to manifest itself gradually as
speed is increased. Up to a Mach number of 0.4, the effect of compressibility
on the aerodynamic characteristics of the wing is only slight and may in
practice be ignored. With a further increase in speed, this effect becomes
more and more noticeable and can no longer be ignored. Starting at flight
speeds of 600 - 700 km/hr and above, drag increases sharply because of
compressibility. This occurs due to the fact that local speeds of the motion
of the air over the wing and at points where the wing attaches to the fuselage
substantially surpass the flight speed. In flowing around the convex surface
of the wing, for example, the air streams are compressed and their
cross-section decreases. However, because the span across the stream must
remain constant, the speed in it is increased. At any sufficiently high flight
speed, the local air speed at any point on the wing or other point on the
structure comes to equal the local speed of sound (Fig. 11).
Figure 11. The Formation of the Local Speed of Sound in
Flow around a Prof i le.
The flight speed at which the local speed of sound will appear anywhere
on the wing is called the critical flight speed V , while its corresponding
Mach number is called the critical Mach number M
Higher values for the
local speeds are observed on the upper air foil profile. As the speed of the
incident flow or the flight speed increases, the local speed reaches the speed
of sound fastest at this point.
Let us examine the air stream surrounding the profile (Fig. 11). Let
us select two characteristic cross-sections of this stream: the large one I
and the small one II. The local air speeds in section II will be greater than
the local speeds in section I as a result of differences between the areas of
these sections. If we increase the speed of the incident unperturbed flow,
the local speeds increase in both sections, but in section II it is greater
than in section I. This is explained by the fact that as a result of the
increase in speed there is a drop in density which is more intense the faster
the speed of the stream. To retain the steadiness of the mass flow weight
rate of air along the stream, the speed in -section II must increase addition-
ally in order to compensate for the great density drop in this section. At
the threshold, the local speed of the flow of air in section II may come to
equal the local speed of sound.
From this it follows that during flight with speed V , the local speed
of sound is achieved at the narrowest point of the stream. It has been
established theoretically that at this instant the critical pressure drop
forms between section I and II which is equal to p : p-j. = 0.528.
It is well known that if the speed of sound is achieved at the narrowest
part of the stream, the speed increases and becomes supersonic if the stream
continues broadening. Therefore, a fully supersonic zone of flow is formed
down with portion of the profile surface during flight with M > M^x-
The greater the flight speed, the greater the zone of supersonic speed will
be. However, far behind the profile the speed must be the same as the flight
speed. Therefore, at some point on the profile there must develop deceleration
of the air from supersonic to subsonic speed. Such deceleration, as
experience has shown, occurs only with the formation of a compression shock.
§ 12. The Dependence of the Speed of the Gas Flow on the Shape of the /27
Channel. The Laval Nozzle
A means for obtaining supersonic speeds in the motion of the gas was .
developed by the engineer Laval (Switzerland) during his work in the 1880 's
on improving a steam turbine he had invented. Laval obtained a supersonic
flow of vapor as it flowed from a special nozzle.
This nozzle, subsequently called the Laval Nozzle (Fig- H) j is a tiibe
which is first compressed and then expanded. The narrowest section of the
tube is called the critical section. If a vapor or gas is run through such
a nozzle at a slight pressure drop in which the speed of the flow in the
critical section becomes subsonic, in the expanded portion of the nozzle the
speed will drop; in this case the Laval Nozzle operates as a typical Venturi
tube. However, if the difference in pressures at the input to the nozzle and
at its output are sufficiently great, in the critical section the speed of
the flow becomes equal to the local speed of sound. In this case, beyond the
critical section, i.e., in the broadened portion of the nozzle, the speed of
the flow does not decrease but, on the contrary, increases. Thus, it was
observed that in sub- and supersonic flows, the dependence of the speed of
the flow of gases on the shape of the channel is directly opposite.
Subsonic flow accelerates in the compression channel and decelerates in
the expansion portion. In contrast, however, supersonic flow loses its
speed in the compression section, while it increases it in the expansion
Therefore, in Figure 11 we see the appearance of supersonic speed after
the stream has passed through the narrow section (point K) .
However, supersonic speed does not increase along the entire length of
the nozzle; at some point it must decelerate to subsonic speed. And herein
lies the cause for the formation of the compression shock.
§ 13. Laminar and Turbulent Flow of Air
Under the effect of internal friction due to the viscosity of air and
the roughness of the surface of the body around which the flow moves, the
speed of air at this surface becomes equal to zero. Depending on the distance
from the surface, the speed of the flow increases and reaches the speed of
free flow. The layer of air in which there is a change in speed from zero
to the speed of free flow is called the boundary layer.
It is well known that the flow of air in the boundary layer may be
laminar (stratified) when the gas flows without being mixed in the neighboring
layers and turbulent when there is random mixing of gas particles throughout
the volume of the flow. The boundary layer also entails phenomena such as /28
burbling (flow separation), the formation of surface friction drag, aero-
dynamic heating, etc.
The interaction of the boundary layer and the compression shocks results
in the following. If the flow in the boundary layer is laminar (Fig. 12),
an oblique compression shock developes
directly on the airfoil profile. Behind the
shock there is separation and turbulence of
the boundary layer; in the turbulent region
a normal shock developes. In general, the
oblique and normal shocks are combined. When
there is an oblique shock, the intensity of
the normal shock will be substantially less
because the flow approaches it, having already
attenuated its speed somewhat in the oblique
shock, with the result that the drag
decreases. Therefore, laminarized airfoils,
i.e., airfoils with very smooth surfaces, are
suitable in that they offer the least surface
friction drag and wave drag at supercritical
flight Mach numbers.
F i g u re 12.
the Prof i 1e: 1
Supersonic Zones; 2 - Com
press ion Shocks; 3 - Sub-
After the normal compression shock there
begins the so-called wave flow separation,
which is accompanied by a decrease in the local air speed. This in turn
results in a sharp drop in the airfoil lift.
During turbulent flow around an airfoil there is no oblique shock and
only one normal shock. The appearance of local shocks on the airfoil
institutes the so-called shock stall. Part of the kinetic energy in the shock
is transformed into heat which is then irreversibly propagated.
At high flight speeds, the characteristics of the compression shock are
a function of the nature of the boundary layer. Experience has shown that
flow in a boundary layer is usually laminar over a certain portion and then
switches to turbulent.
The position of the transfer points of laminar boundary flow to turbu-
lent depend on the shape of the profile, its thickness, roughness, etc. The
surface of a body in laminar flow experiences less friction and less aero-
dynamic heating at high speeds than does one in a turbulent layer.
The state of the boundary layer is reflected not only in the wing drag,
but in its lifting capacity as well. In the boundary layer a flow separation
arises which determines the critical angle of attack and its corresponding
maximum lift ratio.
§ 14. Pressure Distribution at Sub- and Supercritical Mach Numbers
Pressure distribution along a wing profile under flow conditions is shown
in Figure 13. The arrows represent the values of the differences between the
local and atmospheric pressures
at each point on the profile.
The positive overpressure
(atmospheric pressure less
thaji local) is indicated by
arrows pointing toward the
contour, whereas negative
pressure or rarefaction (atmos-
pheric pressure greater than
local) is shown by arrows point-
ed away from the contour.
3. Diagram of the Pressure
tions along the Airfoil Pro-
- vectoral; b - expressed by
sure coefficient (l - upper
face, 2 - lower surface).
To determine and compute
the force of the evacuation on
those points of the profile at
which pressure measurements
were taken, the profile chord
for a line parallel to the chord
cted, then the measured values for the pressure are plotted at a
scale from points specified along the perpendicular to the chord:
overpressure is usually plotted below and evacuation is plotted above,
ts thus obtained then merge in a smooth curve.
In diagrams used in aerodynamics, normally the pressure coefficients
(Fig. 13b) , which represent the ratio of the overpressure at any given point
on the profile to the velocity head of the turbulent flow are plotted at
points on the profile rather than the overpressure, as follows:
is the absolute pressure at a given point;
is the static pressure in the unperturbed flow, i.e., the
atmospheric pressure at flight altitudes;
is the velocity head in the unperturbed flow, determined
by the flight speed and altitude.
From the above it follows that the pressure coefficient p characterizes
the degree of differentiation (in units of the velocity head) of the local
pressure at any point on the upper and lower profile surfaces from the static
pressure in the unperturbed flow. The coefficient p will be negative if the
local pressure on the_j)rofile is below atmospheric pressure. Consequently,
a negative value for p corresponds to the presence on the profile of rare-
faction, where a positive value indicates an increased pressure.
At small Mach mmibers, the diagram for the pressure distribution for each
angle of attack has its own constant form because the air compressibility has
no effect on the nature of the distribution of the pressure coefficients on the
upper and lower surfaces. At high Mach numbers (0.6 and greater), there is
an increase in the rarefaction in which greater rarefaction arises to a
greater degree. This increase in the rarefaction is explained by the effect
of compressibility — density decreases as speed increases. Consequently,
to maintain the constancy of the speed flow rate around the profile, it must
increase further, which in turn causes a further increase in the rarefaction.
At portions of the profile where the flow around it has its greatest speed,
i.e., where rarefaction is greatest, the affect of compressibility will also
To further increase the speed of the incident flow (above M ), the rare-
faction on the leading edge of the airfoil profile decreases while it increases
sharply at the trailing edge, so that here the flow becomes supersonic and
there is additional rarefaction.
The resultant zone of supersonic speed culminates in a compression shock
behind which the local speeds become subsonic. Such a characteristic in the
change of the local speeds for flow around an airfoil profile qualitatively
changes the situation with respect to pressure rarefaction along the profile
as compared to subcritical flow.
From Figure 14 it is clear that at that point on the profile where the
compression shock formed there
Additional rarefaction ______ ^^ ^ sharp and irregular
pressure increase (i.e., de-
crease of rarefaction) . At
Mach numbers greater than
critical, the increase in
pressure in the leading portion
of the profile and an increase
in rarefaction in the trailing
portion leads to a substantial
increase in the drag co-
efficient. Shocks are normally
manifested on the upper then
lower surface in modem pro-
files at positive angles of
Figure 14. Pressure Distribution Along
the Profi le for Mach Numbers Below
(broken line) and Above (solid line)
the Critical Mach Number M
Let us look at the picture
cr of pressure distribution along
the chord of a symmetrical
profile at a given angle of attack for various Mach numbers^ (Fig. 15). If at
small Mach numbers the values of the pressure coefficient p are small, then
with an increase in the speed of the incident flow the rarefaction on the
upper profile contour increases and the curve of the pressure distribution
is displaced upward. When local supersonic zones and compression shocks are
formed on the profile, i.e., for Mach numbers greater than critical, there
is a zone of flow with V > a. This zone is enclosed by the noimal com-
pression shock. The formation of the shock causes a decrease in the rare-
faction on the upper profile. When there is a further increase in the Mach
number, the region of supersonic speeds broaden and the shock gradually is
displaced to the rear. Decreasing the rarefaction becomes much more
significant. The subsequent increase in the Mach number results in the shock
being formed on the lower surface as well, where the rarefaction becomes
greater. With even higher values for the Mach number, both shocks reach the
trailing edge and the entire profile is surrounded by a supersonic flow.
Figure 15- Representative Picture of the Pressure Dis-
tribution on a Symmetrical Profile (solid line -- upper
surface, broken line -- lower surface).
Examination of the picture of pressure distribution gives proof of the
fact that an increase in the Mach number substantially changes both the
characteristics of the curves of pressure distribution and the moment
characteristics of the wing.
CHAPTER I I /32
AERODYNAMIC CHARACTERISTICS OF THE WING AND AIRCRAFT,
THE EFFECT OF AIR COMPRESSIBILITY.
§ 1. The Dependence of the Coefficient c on the Angle of Attack
The dependence o£ the lift coefficient c on the angle of attack a is
an important aerodynamic characteristic of the wing and the aircraft. The
shape of the wing (for a specific number of profiles) in planform has a
significant effect on the character of the change of the coefficient c for
the airfoil at high angles of attack after the local flow starts to break
away. Turbojet passenger aircraft have swept wings, and it is these which
we shall discuss.
Figure 16 shows a graph for the change of the coefficient c as a
function of the angle a of the airfoil with the sweep angle x = 35°.
According to this graph we may evaluate the lifting ability of the airfoil
and determine the angles of attack at which flight occurs. Depending on the
flight speed and altitude for various flight weights, the required values of
c are determined for horizontal flight.
The performace of an aircraft at high angles of attack, the causes for
flow separation (burble) and other characteristics are also determined and
explained by the dependence of c on a.
At high angles of attack burbling begins which distorts the picture
of the flow and introduces a certain decrease in the mean value of the
expansion above the airfoil, the increase in c slows down, and beyond a /55
certain angle of attack called the critical angle of attack, there is no
longer an increase, but rather a decrease in c .
At high Mach numbers (flight cruising speeds) , analysis of the dependents
c = f (a) must be carried out with allowance made for the affect of compress-
ibility, which changes this characteristic to a certain degree.
In swept airfoils, variations in the coefficient c with respect to the
angle of attack have their own characteristics. As can be seen from Figure
16, at angles of attack from -1° to 10 - 12° (for small Mach numbers),
there is a linear characteristic of increase in c . However, at angles of
attack greater than 10 - 12° the proportionality is eliminated between the
increase in the angle of attack and the increase in c ; in addition.
Figure l6. Graphs for the
Coefficient c for a Swept
Ai rfoi 1 at Smal 1 Mach
Numbers (l - wing with
geometric twist of 3°, 2 -
twistj and the Coefficient
c for the Aircraft as a
Function of the Angle of
the increase in c slows down. This is
due to the onset of burbling. At
angles of attack from 17 to 20°, the
lift coefficient reaches its maximum
The change in the dependents
(a) at this portion is a
function of the shape of the leading
edge of the airfoil. The wings in
passenger aircraft have a blunted
leading edge, so that the change in c
in the zone c
Swept wings (as compared to normal
wings) have lower values for the
coefficient c due to the flow around
the wing at a velocity V j~, which by
creating lift becomes a component of
the speed V (see Figure 33) . When
the speed of the flow around the wing
does not correspond to the flight speed,
there arises a lateral displacement of
the air particles in the boundary layer
which, for the central sections of the
wing, is equivalent to the effect which
is obtained when the boundary layer is
blown away or drawn off (see Chapter V,
§ 8). The separation of air particles
from the upper surface is protracted
to very substantial angles of attack,
and before they are reached there is a
steady increase in the coefficient c
for the central portion of the wing.
Because of the great inclination of
the curve c = f (a) to the horizontal
axis in swept wings (as compared to
normal wings), the increase in c as
the angle of attack is increased by 1° it is less than that for a normal
wing, i.e., less than the gradient of the increase for the lift coefficient.
This also determines the lower lifting ability of swept wings as compared
to normal straight wings .
For swept wings, within the range of angles of attack -1.0° - (10-12)°
(linear flow of the relation c =s f (a) on each degree of increase a) the
coefficient c increases by approximately 0.09
The angle of attack at which the decreased growth of c is encountered
and the characteristic vibrations in aircraft are observed is called the
permissible angle of attack a , while the lift coefficient corresponding
to it is
c (Figure 17)
y per ^ ^ ^
The vibration in the aircraft begins after the
burbling begins at the wing tips and the vortex flow strikes the tail
assembly. On the curve (Figure 17) reflecting the total change in c for
the wing as a function of a, the angle
of attack corresponding to the onset
of vibration is determined through the
start of local flow separation at the
wing tip (in the figure, this corres-
ponds to the point where Curve 2 begins
to deviate from the straight line) . When
c is reached by the wing tips , in
y max •' & i- j
spite of the subsequent sharp decrease
in c at these tips, c for the entire
y ^ y
wing begins to increase as the angle of
attack does, although slower than
at the beginning of separation. The
increase in c_ takes place due to the
C ., (C •)
yvib '^ yper
vib , per ■'
Figure I/. The Coefficient c
for Various Parts of a Swept
Wing as a Function of the
Angle of Attack: 1 - central
portion; 2 - wing tip; 3 -
wi ng as a whoJe.
separation-free flow at the central
portion of the wing which occurs at
high angles of attack. For high Mach
numbers, the critical angle of attack
may reach 30-35°.
The aircraft's moving into the
vibration zone indicates that low
speeds have been attained, and in this
case the vibration is a warning for the
In the zone of high angles of attack, there is a smooth change in c
especially close to its maximum. As a result of this, in the shift to
supercritical angles of attack, swept wings have less of a tendency
toward autorotation than do straight wings. In general, the swept wings
on transport aircraft have less of a tendency toward spin.
Because of geometric twist, the running value of the coefficient c for
the characteristic angles of attack during takeoff, climb, horizontal flight,
etc., decreases. As can be seen from Figure 16, for the same angle of attack
a^ , the wing's lift without geometric twist is better, and c _ > c ^. This is
why flight in aircraft with wings having geometric twist is performed at
greater angles of attack than with wings without this twist.
§ 2. The Effect of the Mach Number on the Behavior of the Dependence c = f(a)
Air compressibility affects the dependence of the coefficient c on the
angle of attack. Because of compressibility, an increase in the flight Mach
number of more than 0.4 - 0.5 is accompanied by a qualitative change in the
character of flow around the wing, because the speed of the flow on the wing
increases, as a result of which for one and the same angle of attack the /36
coefficient c increases, i.e., there is an improvement in the lifting
capability of the wing. This is clear from Figure 18 (in which, for example
purposes, the angle a = 4.5° has been selected). The angle of attack at which
vibration begins decreases with an increase in the Mach number, because the
vibration and the flow separation begins sooner than at low Mach numbers.
Therefore, the value c ., also decreases
v-' 'i- per
with an increase in the Mach number. For
example, at M = 0.65, the coefficient
c ., = 0.99, while at M = 0.85 it will
equal 0.52 (Figure 19). In addition,
c also decreases sharply. If from
y max ^ ^
M = 0.65 the coefficient c ., differs
slightly from c , then at M = 0.85
" ^ y max'
the value c ., will be substantially
y vib •'
less than c . Flight accompanied by
y max " r /
vibration usually precedes the onset of
instability in the aircraft with respect
to overload, while at certain values
greater than c , the vibrations can lead
to stalling at certain Mach numbers.
Therefore the value c at which vibration
begins is vital for flight purposes.
If for M = 0.4 - 0.5 the angle of
attack for the onset of vibration (see
Figure 19) equals 12-13°, then for M =
= 0.8 - 0.9 it decreases to 5-7°, and
c ., also decreases. This is especially
y vib ^ ^
dangerous at high Mach numbers because at
the same time as the onset of vibrations, stalling may set in.
Figure l8. The Affect of the
Mach Number on the Dependence
c = f(a): - - - wind-tunnel
tests; flight tests.
Figure 19. The Dependence of a ., and c ., on the
^ ^ vib yvib
In the event that the shift to higher c is not accompanied by the
characteristic vibration (of individual sections of the wing) , to forewarn
the pilot that this shift has occurred, special tubulence sensors are
attached to the wings. They trap the local flow separations on the wing and
transmit the vibration to the control wheel. This, for example, is what was
done on the British turbojet Comet, on which the sensors are set symmetrically
on the leading edge of the center section of the wing (Figure 20) . On the
pilot's instrument panel there is a special
instrument which signals the pilot ahead of
time (before c ., has been reached) that the
aircraft is shifting toward this regime (see
Chapter XI, § 15) .
§ 3. The Permissible Coefficient c and
i ts Dependence on the Mach Number
Flight safety is achieved in turbojet
aircraft at high altitudes and Mach numbers
through restricting the increase in the lift
coefficient by the determined permissible
values of c . This is necessary to
maintain longitudinal stability in the air-
craft. Horizontal flight must be performed
at an altitude and speed in which the value
c , does not exceed c for a normal-
y hor y per
The value c is selected such that it is
., or matches it (Figure 18). From Figure 21
it can be seen that, for example, for a Mach number of 0.65 the coefficient
c =0.86, for M = 0.80 it equals 0.635, etc. The less the degree of
Figure 20. Positioning of
Sensors on the Wing of the
Comet Ai rcraf t.
ized vertical wind separation,
always somewhat less than c
0,2 0,3 D,h 0,5
Figure 21. The Coefficient
c as a Function of the
Mach Number (angle of sweep
X = 35°): -.-.-.- first-
sweep of the airfoil, the greater the value
c . Careful selection of the profiles
permits improving the conditions for flow
aroimd the wing and yields higher values of
Such selection of profiles is especially
characteristic of second- generation turbo-
With high values for the Mach number,
the coefficient c decreases to almost
half its value, and at M = 0.85 it reaches
as low as 0.54. In the zone of small Mach
numbers fup to 0.46), a value of c =
^ y per
= 1.12 -1.2 is used, which permits deter-
mination of the lowest permissible speed
for an aircraft with smooth wings (wing
flaps retracted) .
Further, in examining horizontal flight and the stability and handiness
of the aircraft, we shall return to c and, in addition, we shall consider
a and its representative values,
§ k. Dependence of the Coefficient c on the Mach Number for Flight at a
Constant Angle of Attack
In examining the effect of air compressibility on the lifting properties
of the airfoil in § 2, we noted that for a constant (flight value) angle of
attack, each Mach number is matched by a specific value of c .
As can be seen from Figure 22 (the curve for a = 4.5°), the coefficient
c increases constantly up to a value of M = 0.83, and then decreases. The
reason for such a change in c is due to the effect of air compressibility
on the pressure distribution along the profile (see Figure 9). Even with a
Mach ntimber of 0.4 in the vein flowing over the profile, an increase in
velocity is accompanied by a marked decrease in air density, which leads to
an additional increase in the expansion above the upper surface (§ 10 of
Chapter I). On the lov/er surface, the affect of air compressibility for
these Mach numbers has a lesser effect, so that initially there is an
increase in the coefficient c . During the formation of a compression
shock, the lifting capability of the airfoil decreases. Shock-induction
separation leads to a decrease in expansion on the upper portion of the
airfoil profile, and c decreases. At a given Mach number, when there is a
shock on the lower surface as well, it begins moving back, at first slowly
Figure 22. The Effect of Air Compressi
biiity on the Coefficient c at a
Constant Angle of Attack: 1,2 - swept
wing with geometric twist; 3 " non-
as a whole.
and then rather rapidly. As a
result, on the lower surface the
expansion zone will increase as
the result o£ which the lift and,
consequently, c as well will
start to decrease. Later, as a
given Mach number, the shock on
the upper surface will also start
to move back faster and faster,
which will entail an increase in
the expansion zone and the
coefficient c . The values of
the Mach number at which we
observe the initial increase in
c and its subsequent drop and
renewed increase ("spoon")
depend on the angle of attack
for the profile and the airfoil
As can be seen from Figure 22, for smaller angles of attack
(2-3°), the flow c is smoother with respect to the Mach number and the
"spoon" is only slightly expressed.
This feature of the change in c with respect to the Mach number -- the
"spoon" -- explains the "inverse reaction" of an aircraft (in banking) to
declination in the control wheel (Chapter XI, § 22).
§ 5. The Affect of the Mach Number on the Coefficient c
Let us analyze the formula for drag
Q = c,S
where S is the wing area.
If the angle of attack a is maintained constant, at small Mach numbers
drag will vary proportionately to the square of the speed, while the drag /39
coefficient c at these Mach numbers will be practically independent of speed
and will vary only with respect to the angle of attack. As we can see from
Figure 16, for a = 6-8° the coefficient c = 0.038 - 0.05 (at small altitudes
and speeds). However, the dependence of c on only the angle of attack is
observed at speeds at which the effect of air compressibility may be ignored.
With an increase in flight speed, however, when compressibility does start
to have an effect, the coefficient c increases, and more substantially the
faster the shock stall on the profile developes. The relationship between the
development of the shock stall and the increase in the coefficient c may be
considered from Figure 23. Under Mach = 0.7, the coefficient c is practically-
changeless. After the
flight (flow) Mach niimber
exceeds its critical
value, local compression
shocks begin forming on
the wing, wave drag
appears , and a sharp
increase in the curve c
begins. This makes it
clear that the greater
the airfoil angle of
attack (or the greater
the flight c ) , the lower
the critical value for the
Mach number. With an
increase in the Mach
number, the compression
shocks are displaced
toward the trailing edge
and become more powerful.
At Mach = 1.1 - 1.15, a normal shock appears in front and shocks appear on
both the top and bottom of the trailing portion of the profile.
It must be noted that an understanding of the critical Mach number, as
related to the appearance of the local speed of sound at any point on a swept
wing, has less of a practical value than it does for a straight wing. In
general, the appearance of the local speed of sound on straight and swept
wings does not immediately have a significant effect on the aerodynamic
properties, and will not be noticed by the pilot.
The critical Mach number for a swept wing and the aircraft as a whole /40
is usually related to changes in the total aerodynamic characteristics and this
is understood to mean that flight Mach number at which the pilot becomes aware
of the effect of air compressibility on the handling qualities of his air-
craft, i.e., changes in the stability and handiness. The critical Mach number
Figure 23. Dependence of
on the Mach Number for a
as determined from these conditions is M
0.82 - 0.88. At such a Mach
number, aircraft instability in terms of speed developes (the "spoon" on the
balance curve) and the reverse reaction (in terms of banking) to declination
of the rudder also appears.
In flight practice, concepts are lised such as the so-called limiting Mach
number, which the pilot must know absolutely. It is usually equal to 0.86 -
0.9. This Mach number can reasonably safely be substituted for the critical
Mach numbers discussed earlier.
It should be pointed out that in aerodynamic calculations, the critical
Mach number is sometimes taken to be a flight Mach number whose increase by
0.01 leads to a 1% increase in the aircraft's coefficient c . According to
the latest formulas, the Mach number M =0.78-0.80 for cruising values
' cr ^
c = 0.25 - 0.30. For c = 0.35 - 0.5 at ceiling altitudes, depending on the
takeoff weight the value M decreases 0.70 - 0.74.
As was stated above, when the Mach number is increased above M , a large
supersonic zone of flow appears on the profile, the compression shock is moved
back and expansion in the tail portion of the profile is increased and
initiates an increase in the coefficient c . For non-swept wings, for example,
this phenomenon occurs at Mach numbers 0.04 =0.1 below M
For a further increase in the Mach number above the critical value, the
coefficient c increases as a result of the increase in the local speeds on
the lower profile surface, where a compression shock is also formed. A more
intense increase in c in non-swept wings occurs in the range of Mach numbers
from M to M = 1; with a shift beyond M = 1, however, the coefficient c^
usually decreases. For swept wings, the maximum, value of c corresponds to
the Mach number M = 1.1 - 1.15.
It is known that wing drag is compounded from the profile drag and
the induced drag Q. ; the formation of compression shocks on the wing " adds
the wave drag to these. With respect to this, the inverted form of the
formula for the drag coefficient will be the following:
c = c + c . + c ,
X Xp XI xw
where c^ is the coefficient of profile drag for zero lift, and is compiled
" from the drag of the air friction on the wing surface and the
drag caused by the difference between air pressures on the leading
and trailing portions of the wing. The profile drag for the wing /41
at small Mach numbers can best be established from friction whose
value is only slightly dependent on the angle of attack*; at high
angles of attack the separation drag is added to the friction drag
and the coefficient increases sharply: c = c j- . + c ;
^ ^ xp X fric X pres '
c . is the coefficient of induced drag, which is a function of the
wing lift; it is directly proportional to the square of the lift
coefficient and inversely proportional to the wing aspect ratio:
c . = -^ (here X = — ^ - wing aspect ratio, 1 - span, and S - Wing area);
XI TTa O
* A. p. Mel'nikov. High-Speed Aerodynamics (Aerodinamika bol'shikh skorostey) ,
c is the wave drag coefficient,
Induced and wave drag are by nature pressure drags. IVhen wave drag
developes, the coefficient c increases 3-6 times for straight wings and 40-
70% for swept wings as compared to its values for slow speeds.
Thus, the onset of compression shocks leads to an intense increase in the
coefficient c^ because wave drag is added to the normal profile drag and
§ 6. Wing Wave Drag
It was established earlier that an increase in the flight speed above
critical leads to the appearance of a new, additional form of drag called
profile wave drag.
To explain the nature of this drag, let us once more examine the picture
of the pressure distribution along the upper wing surface for subsonic flow at
sub- and supercritical flight speeds (Figure 14 and 24). As can be seen,
in Figure 24 one section
of the expansion vectors
sort of "draw" the pro- /42
file forward, while the
other draws it back. To
evaluate what would happen
to the wing under the
affect of these "pulling"
forces, all expansion
vectors must be pro-
jected in the direction
of flight. When this is
done we see that at sub-
critical speeds the forces "pulling" forward are negligibly less than those
"pulling" back (Figure 24a). With an increase to supercritical speeds, the
pressure distribution picture changes (Figure 24b), as a result of which the
forces "pulling" the profile forward decrease (expansion becomes less at the
bow of the profile) while the forces "pulling" back increase (because expansion
on the trailing slope of the profile increases by an absolute value) . From
the figure it is clear that the difference in the projections of the vectors
of the "pulling" forces directed to the rear increases, causing an increase
in drag. However, because the extent of the supersonic zones over and under
the wing increases as flight speed increases, there is an even greater
displacement of the largest expansion toward the rear and the trailing edge.
The forces "pulling" the profile forward increase at the same time the pressure
on the leading edge of the profile increases. To sum up, the wing drag
continues to increase. Thus, the wave drag is by nature a pressure drag
because it is dependent on the increase in the pressure difference in front
of the wing and behind it.
Therefore, in aerodynamics wave drag has come to mean the additional drag
Figure 2k. Examples of Wave Drag.
caiised by an increase in the pressure differences in front of the wing and
behind it when there are supersonic zones of flow and compression shocks on the
This drag is called the wave drag because the process of the development
of supersonic zones of flow is accompanied by the development of shock waves
or compression shocks.
From the energetic viewpoint, wave resistance is the result of the
deceleration of air flows on the compression shocks. When this occurs, the
kinetic energy of the flow is irreversibly consumed in heating the air in the
As can be seen from Figure 25b, in the range of cruising flight Mach
numbers, the value of the wave drag c
0.004 - 0.012 or for the mean value
c = 0.025, it will equal 25 - 50% (for aircraft).
At supersonic flight speeds (Mach > 1 - 1.2, Figure 25a), air deceleration
on the bow and tail compression shocks decreases because the angles of
inclination of these shocks decrease, which means that the wave drag itself
At supercritical Mach numbers, aircraft drag increases intensely because
it is a function of both c and V^. From the same figure we see that at a
constant angle of attack, the drag force below M = 0.5 increases as a parabola, /45
while beyond this Mach number this lull does not hold, and the curve deviates
from the square parabola, which is the result of the effect of compressibility
and the development of compression shock.
Figure 25. Dependence of the Coefficient c on the Mach
Number (a) and the Effect of the Relative Profile Thick-
ness on Ac.... for the Wing (b) .
A. P. Mel'nikov. High-Speed Aerodymamics (Aerodinamika bol'shikh skorostey) ,
§ 7- Interference
The increase in aircraft flight speeds has led to an increase in the
importance of interference, i.e., the combined effect of various parts of
the aircraft such as the wing and the fuselage. Usually interference leads
to an substantial increase in drag, especially in the zone of transonic
It has been experimentally established that "positive" interference can
be achieved. This is the interference which aids in decreasing the additional
drag resulting from the points where the various aircraft components are
joined. Turbojet passenger aircraft are basically low-wing aircraft. When
the wing and fuselage are joined in this way, the use of fairings helps to
smooth the junction point of the wing and fuselage to a certain degree.
Positioning the engines in the base of the wing (see Chapter IV, § 8) as was
done on the Tu-104, Tu-124 and Comet aircraft creates an ejector effect — an
"active fairing" -- at the junction point for operating engines.*
Another way of decreasing the drag is using the "rule of area," which
is also applicable for subsonic aircraft.
With respect to this rule, drag in flight vehicles proves to be minimal
when the law of variations in cross-sections with respect to length corres-
ponds to the law of variations in cross-sections with respect to the length
of a body of revolution of least drag. It is well known that drag from the /44
combination of the wing and fuselage (and other parts of the flight vehicle)
will be the same as equivalent drag, i.e., drag having the same law for
variations in cross-section with respect to length of a body of revolution.
Therefore minimal drag may be achieved through decreasing the cross-section
of the fuselage ("squeezing"), at the point where it joins the wing, by a
value equal to the area of the corresponding wing cross-sections (Figure 26) .
dy with win
Figure 26. Examples of the Use of the "Area Law": a
"fuselage - wing" combination without allowance for
the area law; b and c - the same combination with
allowance for the "area law."
S.M. Yeger. Designing Passenger Jet Aircraft (Proyektirovaniye passazhir-
skikh reaktivnykh samoletov) . Mashinostroyeniye, 1964.
The "area law" is also applicable to the junction o£ engine nacelles,
externally suspended fuel tanks and other aircraft components. Thus, for
example, on the Tu-104 and Tu-124 aircraft having wings with a relatively
high wing aspect ratio, the wing and fuselage interference is somewhat
decreased by the substantial distance of the wing tips from the fuselage;
as a result, instead of thickening the fuselage behind the wing, drop-shaped
nacelles are installed on the wing. This yields a smoother change in the
volume of the aircraft along its length without modifying the fuselage.
On the Convair 990, there are four nacelles which are used to carry fuel.
As a result this aircraft has achieved a maximum cruising Mach number of
It is felt that allowance for the "area law" in designing aircraft can
improve their flight qualities by 20-25%. In some cases, however, observance
of this law has proven unsuitable due to complications and difficulties in
designing the fuselage which have resulted in the need for curvature of its
§ 8. The Aircraft Polar. The Effect of the Landing Gear and Wing
Mechanization on the Polar
The polar of an aircraft serves in evaluating the aircraft's aerodynamics.
It offers a graphic representation of the values of the coefficients c and
c at various angles of attack, as well as indicating their variations when
these angles change.
Figure 27 shows the polars of one aircraft obtained as the result of wind /45
tunnel testing and refined with respect to data from flight testing. Let us
determine the characteristic angles of attack and their corresponding aero-
dynamic parameters. The point of intersection of the polar a with the axis
of the abscissa is determined by the zero-lift angle of attack a„ = 1° and
its corresponding coefficient c „ = 0.018 (for a relative airfoil profile
thickness of c" = 10 - 12%); for c" = 12 - 15% the coefficient "c^q = 0.021 -
0.023. The small value for c is obtained through the creation of a well
streamlined shape for the aircraft with a small center section for the
fuselage and engine nacelles.
The aerodynamic tests as to the degree of refinement in the aircraft is
its efficiency. Modem aircraft have a maximum efficiency of K = 15 - 18 at
the optimum angle of attack of 5-7° and Mach numbers of M < 0.5. An air-
craft's lift drag ratio increase with an increase in the angle of attack from /46
a. to the optimal a ^, because at this point c increases faster than c .
^ opt y ^
Starting with an angle of 5-7°, the coefficient c increases more rapidly
(due to the increase in the induced drag) and therefore the performance drops.
Later it will be shown that a is the division point between two flight
regimes: the first and the
second. For the polar a (see
Figure 27), a^^^ = 7° at c^ =
0.55, while K = 17.2.
When the landing gear is
lowered, the polar moves to
the right (polar b in Figure
27) because the coefficient
c increases to the value
Ac ^ . After the landing gear
is retracted, the well doors
are normally closed so that
Ac , = 0.015 - 0.020 and the
lifting ability of the wing
does not change. As a result
the setting for the angle of
attack for polar b remains
the same as for polar a. The
maximum performance for an
aircraft with landing gear
extended decreases in our case
to 12, while a ^ increases to
When the landing gear and
wing flaps are extended (in
landing configuration) the
polar moves to the right and
upward (polar c in Figure 27) ,
and the coefficient c increases throughout the range of angles of attack, the
zero- lift angle of attack becomes negative (a„ = -6°), and the maximum per-
formance of the aircraft decreases as a result of the fact that the coefficient
c increases to a greater degree than the coefficient c .
When the wing flaps are in the takeoff configuration, the maximum per-
formance (landing gear down) decreases to 10-12 (Figure 65).
In gliding toward the landing with landing gear and wing flaps down in
the landing configuration, the performance decreases to 7-8. Extending the
air brake moves the graph of the polar to the right, as the result of which
the performance decreases substantially, particularly in gliding at angles
of attack of 2-3°, at which the landing run is made. Displacing the hinged
flap spoilers causes a sharper drop in the aircraft performance (see Figure
Figure 27. Aircraft Polars: a - landing
gear and wing flaps withdrawn; b - landing
gear down; c - landing gear and wing flaps
extended in landing configuration.
§ 9. The Effect of the Mach Number on the Aircraft Polar
For each flight Mach number we may construct a polar by determining for
this value c and c with an allowance made for the effect of compressibility
and thereby obtain the polar net (Figure 28a) . Earlier it was established
that at subcritical flight speeds the wing coefficient c is almost invariable,
while the lift coefficient c increases starting at M = 0.5 - 0.6. Therefore,
with an increase in the Mach number to M , the polar is pulled forward
because of the increase in Cy and in the region of high angles of attack is
simultaneously shifted to the right due to the increase in c as a result of
an increase in the induced drag._ This is clearly shown in polars for Mach
numbers 0.8 and 0.84 (wing with c = 12 - 15%).
As is well known, aerodynamic performance /47
At super- critical flight speeds at which the wave drag increases substantially,
for a specific Mahh number the polar moves to the right and increases the
shift to that side (in Figure 28a, this corresponds to Mach number of M = 0.84)
as a result of a decrease
Figure 28, Aircraft Polars and Dependence
of Aerodynamics Performance K on Mach
If, however, the
Mach number is so great
that there is wave drag
at almost every angle of
attack, this Mach number
(for any c ) has an
increased value of c and
the polar proves to be
only shifted to the right
(in Figure 28a, the polar
for the Mach number 0.9).
This bears witness to the
decrease in the maximum
performance of the air-
craft, as can be seen in
the figure, in which are
given the tangents to the
polars and the angles for
performance Q„ > G, .
In arranging the polar net, we may construct a graph for the dependence
of performance on c for various Mach numbers (Figure 28b)
performance is obtained for values of c
which are 20-30% greater than the
value for c in horizontal flight
K = 15-17, then at M = 0.8 it will equal approximately 12-14.5
seen from Figure 29, for Mach numbers M = 0.8
If at M < 0.5 the maximum performance
As can be
Figure 29. Maximum Aerodynamic
Performance as a Function of
Mach Number: first-
various second-generation air-
0.84, K = 12-14 and only
at high Mach numbers does it decrease to
11-12. High aerodynamic performance in
an aircraft has a favorable effect on the
volume of fuel consumed per kilometer.
The affect of wing sweep is that with /48
an increase in the angle of sweep, the
aerodynamic performance decreases at low
flight speeds and increases at high
flight speeds . The parameters for
second- generation aircraft wings at
cruising Mach numbers of M = 0.8 - 0.85
have been selected such that K = 13-14
is achieved (Figure 29).
It is well known that for each Mach
number, a high-speed aircraft has its
own relation between the coefficient c
introduce into the polar network values of c
If for various Mach numbers we
for horizontal flight (for
specific weight and altitude) and then join these points, we obtain the polar
for horizontal flight regimes (the dot- and dash line in Figure 28a), which
establishes a relationship between c , c , the Mach number and the horizontal
flight altitude. It is clear from the picture that this polar intersects all
the working polars for Mach numbers from 0.5 to 0.84. The higher the Mach
number, the lower the c at which this intersection occurs. In other words,
the higher the flight Mach number, the lower the value of c^^ required for
CHAPTER I I I
SOME FEATURES OF WING CONSTRUCTION
§1. Means of Increasing the Critical Macii Number
The increase in drag as the Mach number M is raised is an unusual barr-
ier which makes it difficult to achieve high flight speeds. Therefore, tests
have been run on aerodynamic shapes of aircraft at which the shock stall would
begin at the highest possible flight Mach number and would be maintained as
long as possible smoothly, i.e., so that means of increasing the critical Mach
niomber for the profile could be achieved.
The critical Mach number for the profile may be determined according to
the following empirical formula:
where c is the relative thickness of the profile;
c is the lift coefficient for the angle of attack under consideration.
Let us bear in mind that the characteristic parameters for the airfoil
profile are (Figure 30) :
relative thickness c% - the ratio of the maximum profile thickness c
to the chord b;
the position of the maximum profile thickness x % - the relative distance
of the maximtim profile thickness x from the nose to the chord b;
the relative profile curvature f% - the ratio of maximum buckle f to the
the distance from the profile nose to the po^int of maximum profile curv-
ature X., expressed in units of the chord b, - x_%.
Let us examine the effect of each of these parameters on the M number.
The effect of c. The profile thickness has a distinct effect on the value
of the drag. The greater it is, the greater the degree to which the air stream
surrounding the profile is compressed, and consequently the sooner the shock
stall will occur at lower Mach numbers. In contrast, decreasing the profile
thickness displaces the moment when the shock stall occurs to a higher Mach
number. Figure 31 gives a clear example of the degree to which the thinness
of the profile results in a greater critical Mach nxomber M
f^ Separation lin^)
Figure 30. Geometric Parameters and Shapes of an Air-
foil Profile: a - profile with positive curvature; b -
symmetrical profile; c - "inverted" profile with nega-
tive curvature (Douglas DC-8) .
!!,SS OJ Q,7S 0,6 I,IS on n
Figure 31. The Effect of Air-
foil Profile Thickness on the
Coefficient c for Various Mach
Aircraft wings carry fuel, with the
result that the relative profile thickness
is 10 to 15%. This is necessary to obtain
sufficient volume and maintain wing
As an example, let us determine the /50
critical Mach number for profiles with
relative thicknesses of 10 and 15% if
c_ = 0.3. Calculations show that for
c = 10%, M
= 1 - 0.7/07l0 - 3.2
while for c = 15% M
1 - 0.7 / c - 3.2CC
0.1 • 0.3-'-'^ = 0.722,
1 - 0.7/0715" -
- 3.2-0.15 -• 0.3-^*^ = 0.651. As we can
see from this example, the lower the
relative profile thickness, the greater
the critical Mach nimiber.
When there is a change in the angle of attack, and consequently the value
c (for example, let us take c = 0.4 and c = 10%), we obtain a different
value for the critical Mach nimiber M:M = 1 - 0. 7/0. 10 - 3.2 ' 0.10 • 0.4 =
= 0.691. Thus, an increaise in the Mach number (c ) has led to a decrease in
M^^ from 0.722 to 0.691. This is explained by the fact that as the angle of
attack increases, the upper air stream is compressed stronger by the profile.
Thp straight-away sections in the stream decrease more intensely, as a
result the velocity increases more sharply, and the speed of sound is
attained at a lower Mach flight number. This is why an increase in the flight
altitude (an increase in c ) decreases the critical Mach number.
Second-generation aircraft have airfoil profiles from c^ = 10-12%, which
makes it possible to increase the cruising Mach flight number to 0.8 - 0.85
without a substantial increase in drag. Usually the optimum cruising flight
speed corresponds to M or less.
The effect of a positive maximum thickness and the relative profile
curvature. It has been experimentally established that with identical
wing thicknesses, the profile which has a higher critical Mach number M is
the one in which the maximum thickness is closer to the center, i.e. , for
X = 35-50%. This is explained by the fact that with such a value for x ,
there is a smoother profile contour, and consequently a smoother change in
pressure and velocity along it (Figure 32) .
A decrease in the profile
curvature has a favorable effect
on the aerodynamic characteristics
at high flight speeds . A
symmetrica^l profile (Figure 30, b),
in which f = 0, other conditions
being the same, as a higher
critical Mach number. However, in
such profiles the values for c
^ y max
are small (by comparison with
asymmetric profiles), so that their
use on transport aircraft is
difficult. Recent years have shown
a broader use of the so-called
"inverted" profile, i.e., a
profile having negative curvature
(Figure 30, c). These profiles,
usually used in the basic section
of the airfoil, satisfactorily
solve the problem of the highly complex interference between the wing and the
fuselage, creating smooth flow. The physical nature of the effect of relative
curvature on the value M is the same as the effect of the thickness.
Decreasing the maximum profile thickness, shifting it to the middle of /51
the chord, and decreasing the profile curvature all increase the value of
the critical Mach number by a total of 0.02 - 0.06.
Figure 32. Effect of the Position of the
Maximum Airfoil Profile Thickness on the
Critical Mach Number M
without rarefaction peak; b
with rarefaction peak.
a - prof i le
prof i le
The effect of wing sweep. The optimum effect in increasing the critical
Mach number is achieved through the use of swept wings.
As wing sweep increases to 35°, the critical Mach number increases by
0.07 - 0.08 as compared with the critical Mach number for a straight wing or
profile. Let us see how this is achieved.
The lift of the wing and the tail assembly is determined by the value of
the aerodynamic force of the pressures arising as a result of changes in the
local flow velocities induced by the external contours of the profile across
the entire wingspan or tail span.
Let lis expand the flight speed V over two components: one, perpen-
dicular to the leading edge* of the wing -- V j., and the other directed along
the leading edge of the wing -- V^ (Figure 33, a). The component V^ (effective
speed) determines the value of the local speeds and expansions along the pro-
file, and consequently the value of the lift as well. The component V, is
not involved in the creation of the aerodynamic pressure forces. It does have
an effect on the boundary layer and, consequently, on the flow separation.
In conjimction with the fact that V -. is always lower than V , the local
speed of sound will be achieved later and, consequently, the critical Mach
number will be greater. The shock stall on the profile will set in at a
higher flight speed. This means that the critical Mach number in swept wings
will always be greater than in straight wings or the profile.
The critical Mach number for a swept wing, with allowance made for the
effect of flow characteristics on the pressure distribution along the span, /52
may be determined from the formula:
M = M
crx cr.prof 1 + cos x '
where x is the angle of sweep for the wing.
For wings having a sweep of 35° (cos 35° = 0.82), the formula assumes the
following form: M _.^o = 1.1 M -. For example, for a relative profile
^ crx "35 cr.prof ^ ^
thickness of 10%, we obtain a Mach number M „_o = 0.795. We must bear in
mind that the empirical formula for determining the critical Mach number
offers an error of 15-20%.
Along its span, the aircraft wing has changing values relative to the
thickness. Therefore, the critical Mach number also has various values.
The effect of wing sweep, by increasing the critical Mach number, is
decreased at the point where the central section of the wing joins the
fxiselage. Here the wing is subjected not to oblique airflow (resulting from
decomposition of the incident flow into two components) , but to straight air-
flow. The critical Mach number is increased through increasing the sweep of
the central portion of the wing along the leading edge. Thus, if the angle
X = 30-35°, in the central section of the wing it reaches 40-45°, i.e., the
wing is given a "crescent" shape in planform. The Tu-104 and Tu-124 air- /53
craft have a slightly expressed "crescent" shape.
Strictly speaking, V j, is perpendicular to the aerodynamic center line MN,
and the component V, is directed along this line, because the wing is
looked upon as tapering. Our allowance has been made for simplicity in
V„„„= 1305 km/hr Vs" •'^°° ^""/'^'^
Figure 33. Development of Flight Speed on Swept Wing
and Possible Positions of the Leading Wing dge Relative
to the Mach Cone: 1 - subsonic leading edge -- wing
located within cone (subsonic flow); II - sonic leading
edge (flow at the speed of sound); III - supersonic
leading edge (supersonic flow).
The critical Mach number for the wing in passenger aircraft is below
unity. For clarity in representation, we will show that for a wing with
thin profiles (c = 4-6%), at an angle x = 55-60° the critical Mach number,
determined according to the formula already presented, may be greater than
unity. However, for an isolated profile, as has already been noted, this is
The shock stall in a swept wing occurs later, and not simultaneously
throughout the wingspan, and less intensely than on a straight wing; in
addition, it does not lead to a sharp change in the total aerodynamic
characteristics of the aircraft.
At various points on the wing, the shock stall developes in different
ways. Recent studies have shown that in the center of the wing the shock
stall begins later than at the tips, but because of this increases more
intensely. As a result, the negative effect of the central portion of the
wing is felt not so much in the sense of a decrease in the critical Mach
number as a more rapid increase in the wave drag than at the wing tips,
although it starts to increase sooner on the tips.
There is substantially less wave drag in a swept wing than in a straight
one, which may be clarified thusly.
Let us assume that local compression shocks arising in profiles from
which the wing is shaped start at the line MN (Figure 33, b). In each profile,
the local shock will be normal, while for the whole wing the total shock,
also located along the line MN, will be oblique (with respect to the incident
flow). As has already been stated, the shock stall developes more weakly when
there is an oblique shock.
The shock front is located along the leading edge of a swept wing at the
instant when V j, becomes equal to the local speed of sound. On a wing with a
sweep angle x = 35°, this occurs at a flight Mach number equal to 1.22. Let
us show this.
As can be seen from Figure 33, a, the speed V ^ = V cos 35°. Let us
equate it to the speed of sound: a = V cos 35°, i.e., a = 0.821 V ; then
^ V V P°^ P°^
M = -E — = P ■ ^ - = 1.22. Thus, a wing with x = 35° may be used also for
flights at low supersonic speeds.
As can be seen from Figure 33, c, a Mach cone forms at the tip of the
angle forming the leading wing edge when a swept wing encounters supersonic
flow. This Mach cone assumes the form of an oblique compression shock. If
the leading wing edges lie within the Mach cone, they are called subsonic.
With respect to the degree to which the surface of the Mach cone approaches
the leading edge, the wave drag ratio increases and reaches it highest value /54
at the instant when the leading edges meet the cone surface. When there is
a further increase in the speed, the leading edges of the wing go beyond the
boundary of the Mach cone, after which the surfaces of the Mach cone move
away from the edges. In this case, the leading edges are called supersonic.
Passenger aircraft designed in recent years have an optimum angle
X = 20-35° and a mean relative thickness of 10-12%. The use of a greater
sweep angle (particularly one equal to 45°) is inadvisable in terms of a
weight-drag ratio for the wing because of the onset of torque and, additionally,
because of poorer takeoff and landing conditions caused by a lower value for
Use of a wing with a 35° sweep results in a 10-25% drop in wave drag for
flights at M = 0.80 - 0.85, which substantially decreases the overall drag.
At the same time it becomes possible to maintain the lift-drag ratio for the
aircraft within limits of 13-15. The effect of the sweep angle on the
coefficient c is given in Figure 34.
In addition to the parameters already discussed, the wing aspect ratio X
also has a deteimining effect on the critical Mach number. A substantial
increase in the critical Mach number results for X = 1 - 1.5. In wings with
small aspect ratios (X = 1.5 - 2.5), the critical Mach number is greater than
in wings with high aspect ratios (X = 5-8) . This is explained basically by
the so-called end effect.
1 1 1 1 1 1 . .
y Y without flow
0,2 a;s 1,0 ijt ti
Figure S'*. The Effect of the
Sweep Angle on the Dependence
c = f(M)
Figure 35- The Effect of Airflow Past
the Wing Tips on Pressure Distribution
over the Upper Surface.
During flight, pressure below the wing is greater than above it. There-
fore, there is an overflow of air at the wingtip from the region of greater
pressure toward that of lesser pressure, i.e., a certain pressure balance
takes place, thanks to which the maximum rarefaction over the wing decreases
(Figure 35). The influence of the end effect is substantial only close to the /55
wingtip. If the wing aspect ratio is decreased, the relative length of these
sections increases and the end effect is spread over a large section of the
For passenger aircraft at an angle x = 35°, the optimum A = 6-8; there-
fore the critical Mach number in this case undergoes no change.
§ 2. Features of Flow Around Swept Wings
In the preceding section, which examined the development of the speed
, we simplified the picture of the flow around a swept wing. Actually,
however, this picture assumes a complex spatial scheme. Let us spend some
time discussing the various basic moments. To this end, let us examine air
streams flowing around the middle and end portions of the wing (Figure 36) .
As a result of the spatial character of the flow of the stream as we approach
the center section of the wing, it becomes wider. As a result of the
constant air consumption along the stream, this leads to a decrease in speed
in the center section of the profile, and consequently to a decrease in the
rarefaction over the rising part of the profile in the middle of the wing.
On the descending part there is a constriction of the stream and a consequent
rise in speed and increase in rarefaction. Thus, in the middle section of
the wing the rarefactions decrease on the rising section of the profile, while
they increase on the descending section.
At the tips of swept wings, the picture is reversed. Here the streams
approaching the wing are first constricted, which leads to an increase in
velocities on the rising profile section. As a result, rarefactions on the
leading profile sections increase. As the profile descends, the stream starts
broadening, which leads to a decrease in velocities and rarefaction.
Figure 36. Representative Character
for the Flow of Air Streams in the
Middle and at the Ends of a Swept Wing.
Figure 37. Representative Picture of
Pressure Distribution at Various
Sections along the Wing: 1 - at the
tips; 2 - in the middle of the semi-
span; 3 ~ in the central section.
Figure 37 shows that at the center sections of the wing, the maximian /56
rarefaction is displaced to the rear, whereas at the tip sections, in
contrast, the greatest rarefaction is found at the leading part of the pro-
file. In addition, the value of the rarefaction peak is higher at the tips
than in the center and base sections. Therefore, the tip sections of the
wing are more loaded (have greater lift) than due the base sections.
The observed feature of pressure distribution along the chord of the wing
leads also to another distribution of load along the span (in contrast to
Figure 38 shows the load distribution along the span of swept and
Figure 38. Diagram of Load Distribution
Along the Span of a Swept and a Straight
Wing: -,.- geometric twist; -.- aero-
dynamic twist; flat wing.
straight wings, as well as
changes in the maximum values
of the coefficient c
y sec max
for various wing sections*.
The difference in the
characteristic for the change
in c in straight and
y sec max "
swept wings is explained in
the following manner. The
overflow of air past the wing
tip from the lower to the
upper surface in a straight
wing has an effect only on a
Pashkovskiy, I.M. Characteristics of Stability and Controllability in High-
Speed Aircraft (Osobennosti ustoychivosti i upravlyayemosti skorostnogo
samoleta). Voyenizdat. 1961
small section, as a result of which the value c is identical almost
' y sec max
everywhere on the span and only toward the wing tips does it start to decrease.
In swept wings, however, the decrease in c from the base to the tip
f ^ ' ' y sec max ^
is related not only to the overflow of air past the tip but also with the
nonsimultaneous increase in the flow separation along the span. This
separation is highly dependent on the air overflow in the boundary layer due to
the component V, (see Figure 33, a). Therefore, the end sections of the swept
wing undergo separation before all the others, i.e., they are the first to /57
attain the values c
y sec max
As can be seen from the figure, the end sections of the swept wing
achieve c faster than do the sections of the center and base
y sec max
portions of the wing. In straight wings, on the other hand, c is
'^ ^ & & J y sec max
reached earlier in the center section of the wing.
Therefore, with an increase in the angle of attack the flow separation
reaches the end sections of the swept wing and the center sections of the
straight wing sooner. In addition, the overall end flow separation on the
swept wing facilitates the speed V^ , which causes the boundary layer to move
toward the wing tip and causes it to thicken. The boundary layer seems to be
in a sense sucked from the center section and built up at the ends of the
wing. The "swelling" of the boundary layer and the premature separation
at the wing tips is one of the basic drawbacks of swept wings.
The end flow separation leads to the development of the pitching moment,
which affects the longitudinal stability of the aircraft adversely,
especially at slow flight speeds. Flow separation in the aileron zone leads
to a drop in the laterial handiness.
Along with end flow separation, at low flight speeds (greater than the
angle of attack) , such a separation is possible also at high speeds at low
angles of attack, which is explained by the interaction of compression shocks
with the boundary layer during flight at high altitudes. As in well known,
at high altitudes flight is performed at high angles of attack (to obtain
the necessary value for c , ^ ) . With an increase in the angle of attack, the
value for the critical Mach number decreases. When the angle a increases due
to vertical gusts, compression shocks may form earlier (because the critical
Mach number is low) , which aids in the development of flow separation. In
all these cases, during separation there is the characteristic vibration, and
in some cases there is even pitching down.
Redistribution of load along the span of a swept (in contrast to a
straight) wing always leads to a displacement of the equivalent aerodynamic
force of the wing backward or forward along the chord, and therefore is
accompanied by a change in its longitudinal moment.
As can be seen from Figure 39, when the wing is swept, each section is
displaced relative to each other in such a way that in toto the points of
application of the increasing aerodynamic forces for these sections form a /58
line which is inclined along the perpendicular to the axis of the wing (the
axis oz) by angle x- The distance from the axis oz to the points of
application of the aerodynamic forces for these
sections differ according to span. In straight
wings, on the contrary, the points of appli-
cation of the increasing aerodynamic forces
for the sections lie practically on a straight
line parallel to the axis, i.e., they are
equidistant from the laterial axis of the wing
in all sections across the span. This feature
for the load distribution along the span in
swept wings changes substantially either with
a change in the angle of attack or a change in
the Mach number.
Figure 39. Example of the
Effect of Load Distribution
Along the Span on the
Longitudinal Moment of a
From Figure 40 we see that an increase
in a leads to a greater load on the central
section of the swept wing and a lightening
of its end sections. In this case, the
pressure center for the wing shifts forward along the chord, which creates
a tendency toward pitching. The onset of pitching corresponds to the moment
of the onset of separation, which starts at that section of the wing where
the aileron are located.
If there is a change in the Mach number and a remains constant, there is
also a redistribution of load along the span. This is accompanied by an
imequal development of shock stall on the wing in the process of reaching and
surpassing critical speed. As we can see from Figure 40, an increase in the
flight speed up to critical leads first to a certain loading of the end
sections of the swept wing. Then, with the development of the shock stall
at a Mach number somewhat greater than M , the end sections start losing
their load. The initial increase in the loading of the end section leads to
the development of a slight diving moment, i.e., to a change in the longi-
tudinal stability*. Subsequent changes in the load distribution are brought
about through the propagation of the shock stall along the upper wing surface
to the base and middle sections of the cantilevers, as well as the development
of the stall on the lower wing surface. All this leads to a certain dis-
placement of the wing pressure center (p.c.) forward along the chord and the
appearance of a pitching moment at Mach numbers greater than critical, but
less than unity (sonic speeds).
Distinct changes in the load distribution along the span of a swept
wing may also lead to its flexible deformation (buckling and twisting). In
the event of deformation, the local angles of attack at various points along
Pashkovskiy, I.M. Characteristics of Stability and Controllability in High-
Speed Aircraft (Osobennosti ustoychivosti i upravlyayemosti skorostnogo
samoleta). Voyenizdat. 1961
the wing change dissimilarly, becaiase the degree of these changes is a
function of the aerodynamic forces acting on the wing. These latter, in turn,
are functions of the angle of attack, flight speed and Mach nimiber.
y J^\&ti along wing
Figure 40. Change in the Load
Distribution Along the Span of
a Swept Wing as a Function of
the Angle of Attack and the
Figure ^1. Decrease in Angle of Attacl<
for Bend in a Swept Wing: a - non-
deformed flexural axis; b - flexural
axis of cranked wing.
In the event of buckling of a swept wing (Figure 41) relative to the 0-0
axis, the points 1 ard 3, lying close to this axis, will have less of a
vertical displacement than points 2 and 4. As a result of this, the chords
1-2 and 3-4 are turned relative to the flexural axis by a certain angle, and /59
the entire wing turns to the side of the decrease in the angle of attack.
Thus, for a wing with normal sweep, in the event of twisting induced by
aerodynamic loads directed upward from below, there is always a decrease in
the angle of attack of the wing section the closer this given section is to
the end of the wing. This also aggravates pitching, in that the end sections
have smaller angles of attack and, consequently, lower values for c ,
This fact, along with the forward displacement of the pressure center as the
angle of attack and speed increase, may also lead to aircraft instabilities
within a specific range of Mach numbers.
§ 3. Wing Construction in Turbojet Passenger Aircraft
In designing aircraft for cruising Mach numbers of 0.8 - 0.85, strict
attention must be given to the selection of wing parameters. We are already
familiar with certain parameters, and now we shall continue our examination.
It has been established that for subsonic passenger aircraft, the optimum
parameters are an angle of x = 35° and a wing aspect ratio o£ X = 6-8. With
such values for X, flight distance is substantially increased.
Narrowing the wing in planform n = r is decided through the selection /60
of conditions yielding best stability characteristics and characteristics of
longitudinal stability, so as to eliminate separation flows at the wing tips.
For a 35° sweep, the optimal selection is n = 3.5 - 4.5*.
The remaining wing parameters are selected from calculation of the
optimal lift properties for the wing.
It has been established that the dependence of the coefficient c (as
well as the coefficient for the longitudinal moment m , Figure 140) on the
angle a proceeds linearly to a ., , at which point there are local flow
separations on the wing and this relation is no longer valid. This leads to
the fact that at high angles of attack there is a decrease in longitudinal
stability (in Figure 140, this corresponds to the so-called "balance point").
The disruption in longitudinal stability is quite representative of swept
wings. It is troublesome not only in that it affects the longitudinal
stability of the aircraft adversely, but in addition the flow separation from
the wing tips decreases the effectiveness of the ailerons and asymmetric
separation may result in pitching down.
Therefore, in establishing the aerodynamic arrangement of the swept wings
in passenger aircraft, maximum cruising flight speeds and minimum landing
speeds are achieved through holding the development of the flow separation
to the highest possible angles of attack and the highest Mach mombers. The
following means are used to achieve this.
1. The aerodynamic twist of the wing -- the selection of the wing
design from various profile types, the profiles offering the lowest lift being
at the base of the wing, while those with the greatest lift are at the tips.
This results from the change characteristic for c with respect to the
" y sec max ^
wing dimensions (Figure 38). The selection of profiles with greater lift for
the wing tips (with f =^ 2.5 - 3% and greater) with the reverse positioning
of maximum thickness (x = 35 - 50%) permits a certain increase in c
^ c y sec max
at the wing tips and, at the same time, increasing the angle of attack and
thereby achieving c
' y sec max
Symmetrical profiles (sometimes with slight curvature) or profiles with
negative curvature -- "inverted" profiles -- are positioned at the base of
The DC- 8, Convair 880, the Boeing- 70 7 and the VC-10 have "inverted"
* Yeger, S.M. Design of Passenger Jet Aircraft (Proyektirovaniye passazhir-
skikh reaktivnykh samelotov). Mashinostroyeniye. 1964.
profiles in the center sections of the wing. This has not hindered the overall
lift of the wing and has made it possible to use profiles with c = 12-15%
without a significant increase in c at high flight Mach numbers.
2. Geometrical twist is the gradual spiral effect (positioning at a /61
smaller angle) of the wing tips and middle wing sections relative to the base
at an angle of 2-5° (for example, if the angle is +3° at the wing base, while
it is -1° at the wing tip, the twist angle equals -4°). This changes the
lift distribution along the span toward the side of greater load for the wing
base and unloading for the wing tips. During flight, this type wing may
achieve higher angles of attack (calculated with respect to the chord of the
base profile) before the wing tips reach separation. Figure 16 shows that the
geometrical twist has an affect on the extension of the relation c = f (a) ,
moving it to the right. ^
Having established the geometric twist, we must take into account the
bending and warping of the wing, as shown in Figure 41, so as to not obtain
negative lift at the tips.
It was noted earlier that with geometric twist, the required c ^ is
achieved at a slightly higher flight angle of attack.
3. Positioning aerodynamic baffles 16-20 cm high (an average of 2-4%
of the local wing chord. Figure 42) on the upper wing surface. The baffles
separate the wing into portions and hinder the overflow of air in the boundary
layer along the wing span, resulting in a decrease in the thickness of the
boundary layer in the tip sections. This leads to an increase in the local
values for c in the end sections (by comparison to a wing without
y sec max ^ j f &
baffles) , and consequently aids in holding off the onset of flow separation
in these sections until the high angles of attack.
••TV-^^l ^ r^ \
r ' 4- ^Al / JX\^ \
* » 0,0!-0,D'eb
Figure ^2. Arrangement of Aerodynamic Baffles on Upper Wing Surface:
1 - 1 i ne of Mk chord; 2 - point of onset of flow separation and
burbling; 3 - aileron; k - baffle; 5 - ai r stream (enlarged scale);
6 - vortices separating from wing with baffles; 7 - possible baffle
In the wing section closest to the fxiselage (between the baffles and the
fuselage) there is a thickening of the •boundary layer and a decrease in
c . Lateral flows arise within the limits of only one section,
y sec max ^
vortices form at the baffles, and the boundaay layer flows off with these. /62
Thus, because of the lateral overflow of air in the boundary layer when
the wing is equipped with baffles, the initial flow separation on the wing
section between the baffles and the fuselage is maintained and separation
from the outer section of the baffles and the wing tips is forestalled.
Because the tendency toward separation of the boundary layer weakens, there
is an improvement in the lift distribution along the wing span. The
separation zone is displaced toward the middle sections and, in some indi-
vidual cases, even toward the base of the wing. Aerodynamic baffles have
been installed on the wings of the Tu-104, Tu-124, Tu-134 and Caravelle
A similar effect is created by the pylons which support the engines on
such aircraft as the Boeing- 707, the Douglas DC- 8 and the Convair 880 (see
Figure 65). However, pylons behave basically like baffles on the lower
wing surface, where there is substantially less cross current in the boundary
layer. Only that portion of the pylon which captures the upper wing surface
at its nose has an effect on the wing.
The 11-62 has swept wings with so-called "notches" in the leading edge
(Figure 43) . The "notch" forms a constant vortex cord on the wing surface
which acts in the same manner as an aerodynamic baffle, increasing the build
up of the boundary layer behind itself with the result that it does not
overflow to the wing tip.
There are of course other means for tightening separations from the wing
at low speeds, and they will be discussed in Chapter V, § 8.
The Boeing- 707, the DC-8 and other aircraft tighten the flow through the
use of vortex generators. Their basic purpose is the creation of a system of /63
vortices for activating the boundary layer (Figure 44) .
Figure ^3. Positioning of "Notches" on the Leading
Edge of a Swept Wing.
Figure 44. Positioning of Flow Vortex Generators
on the Wing of the Boeing-707 (h = 10-12 cm,
a = )5°, 1 = 15-30 cm, D = 40-60 cm).
The principle behind the action of vortex generators is based on the
fact that a system of vortices having a parallel influence on the boundary
layer flowing around the wing surface at the upper limit causes an increased
mixing of the boundary layer with the outer flow. Air particles carried from
the outer flow by the vortex displace the particles in the boundary layer and,
through mixing with them, are entrapped in the outer layer. There is intensi-
fication of the boundary layer which restricts its breaking away from the
compression shock. In those instances where break away nevertheless occurs,
the vortex system excited by the vortex generators creates in intermixing
effect in the separated flow as well, as a result of which the flow
separation region is localized and the boundary layer again "adheres" to the
Setting up vortex generators has succeeded in forestalling the development
of flow separation at high angles of attack and flight speeds (an increase in
the critical Mach number to 0.02 - 0.07). Aileron effectiveness increased
because the vortex generators inhibit separation of the boundary layer along
the rupture line of the upper wing surface when the aileron is down. Vortex
generators set in the base section of the wing (Boeing-707) decrease lift at
high angles of attack through flow separation.
In addition, on the Comet-4c there are the so-called sensors (special
plates. Figure 20) which break up the flow at the base section of the wing
at high angles of attack and by so doing decrease the pitching moment.
In summary, the measures described (including those laid out in Chapter /64
V, § 8) make it possible to design aircraft wings with the shape shown in
Figure 45. It must be noted that if along the 1/4 chord line the angle
X = 35°, then along the leading edge the sweep may be somewhat greater (in the
Yeger, S.M. Design of Passenger Jet Aircraft (Proyektirovaniye passachir-
skikh reaktivnykh samelotov) . Mashinostroyeniye. 1964.
figure this corresponds to an angle of x = 41° in the base section of the wing
and X = 38° in the outer wing section) .
Figure 45. Schematic Diagram of Aircraft Wing:
1 - inside spoiler; 2 - inside flap; 3 - outside
spoiler; h - outside flap; 5 - inside aileron;
6 - outside aileron; 7 - flettner trim tabs; 8 -
intermediate ribs; 9 - landing gear pod; 10 -
secondary control surfaces; 11 - tip ribs; 12 -
spar axes; 13 ~ wing stump joint; ]h - wing
j o i n t ax i s ,
Tables 3-5 present the values of parameters (in percentage) for the
following variations in wing aerod/namic arrangement:
a) for a wing without geometric twist ( cruising Mach number M
= 0.75 - 0.78, cj)^.^ = +1°)
At wing stump joint
At wing joint axis
At tip rib
* Relative thickness along flow.
b) for a wing with geometric twist (engines in tail section o£ fuselage, /65
cruising Mach number M
0.8 - 0.82, and <}> -i^ = +1°;
A . ., = -l^SO'):
At wing stump joint
At wing joint axis
At tip rib
* Relative thickness along flow.
c) for wing with geometric twist (engines in tail section of fuselage,
ing Mach number M . = 0.82 - 0.85, 4 ., = +3°, ^. . ., =
^ cruise T^ase vib inter, rib
= -l°^ ■
At wing stump joint
At tip rib
§ k. Drag Propagation Between Separate Parts of Aircraft
Total aircraft drag is known to be the composite of drag in the individual
sections. For various flight speeds (Mach numbers) diverging drag propagations
result between these parts mainly due to the onset of wave drag at the
respective Mach numbers. In subsonic aircraft, around half of the total drag
is created by the wing. Table 6 shows representative values Ac for the basic
aircraft components with the engines set in the tail section of the fuselage
(the data pertain to horizontal flight at a Mach number of M = 0.8, at which
c for the entire aircraft equals 0.0305, while c = 0.4).
X ^ ' y ->
It should be noted that the portion of wave drag for M = 0.8 at c = 0.4
(corresponding roughly to the high angle of attack a ~ 5.5°) is approximately
20% (Ac. -T = 0.006). Having the landing gear down (Ac = 0.015 - 0.020) at
tai 1 X
low flight speeds creates approximately half of the entire aircraft drag.
In % of
Rudder- fin unit
Landing gear pods
Side engine pods
Center engine intake
CHARACTERISTICS OF THE POWER SYSTEM
Jet engines and, in particular, turbojet engines generate high in-flight /66
thrust and, consequently, high thrust horsepower (30,000 - 60,000 hp)
necessary for propelling aircraft weighing 40 - 160 tons at a speed of 850 -
Piston and turboprop engines use up all or almost all the energy from the
fuel in rotating the propeller. It is the propeller which, driven in its
rotation by the engine, creates the thrust. Therefore the propeller is called
the prime mover of the aircraft. The power system for piston and turboprop
engines comprises both the engine and the prime mover, which create the thrust.
In the operation of a jet engine, however, the thrust is achieved in-
directly as the interaction of all the forces acting on the surface of the
engine components. The jet engine organically combines within itself the
engine in the normal concept of the word and the prime mover.
During test-stand operation of modem turbojet engines, the pressure at
the compressor exhaust equals 5-10 atm or more.
The gas temperature at the combustion chamber exhaust is 1,00 - 1,200°
abs. The power generated by the gas turbine is 60,000 - 90,000 hp for engines
with a thrust from 5,000 to 10,000 kG.
As it exists from the turbine, the gas still has a high amount of heat
energy, its pressure is greater than atmospheric, and its temperature equals
800 - 1,000° abs. Through the process of expansion, the thermal energy of
the gas at the' exhaust nozzle is transformed into kinetic energy, and as a
result of the high speed of the gas exhaust, the exhaust thrust is generated.
§ 1. Two-Circuit and Turbofan Engines /67
Attempts by aeronautical engineers to increase engine thrust and decrease
fuel consumption led to the creation of the two-circuit and turbofaxi engines
(Figure 46). Fuel consumption in particular decreased by 15-20% by comparison
with consumption in normal turbojet engines.
The two- circuit (turbofan) engine is a gas turbine engine in which the
excess turbine horsepower, in contrast to the turboprop engine, is transmitted
to a compressor or fan enclosed in the circular cowling.
The two-circuit turbojet engine may assume one of several structural
designs (Figure 46a and b) which are characterized by the existence of an
additional air circuit through which, after compression, part of the air which
has been sucked in is fed to the combustion chamber and turbine bypass directly
to the outlet, thereby increasing the mass and decreasing the speed of the
Two- contour engines in which the volume of air passing through the
supplementary circuit is relatively great while the degree of compression of
this air is small are usually called turbofan engines. At present there are
in use two- circuit engines of this type and turbofan engines, which are derived/68
through the installation of a fan in addition to the normal turbojet engine
(Figure 46c and d) . The expediency of creating turbofan engines based on
series turbojet engines for civilian aircraft is justified through their
great economy and high reliability during use.
Figure k6. Various Types of Two-Circuit and Turbofan
Engines: a - normal scheme (Rolls Royce "Conway" engine);
b - two-circuit engine with air displacement from outer
contour with gases from the inner contour (Rolls Royce
JT8D "Spey") ; c - turbofan scheme with forward fan
(Pratt-Whi tney JT3D) ; d - turbofan with rear fan (General
When a turbojet engine is being designed strictly along the two-circuit
plan, optimal parameters are obtained if the design and the parameters of the
turbofan engine are to a great degree determined and limited by the parameters
of the initial turbojet engine.
Figure 47 shows a simplified schematic of a two-circuit engine. Atmos-
pheric air enters the air scoop through the two layers of blades which form
the fan B. From this fan, which is in effect a low-pressure compressor, the
air moves on in two separate paths. One part of the air moves along the outer
body of the basic engine contour through the second contour C, while the other
part moves through the high-pressure compressor D. From there it moves through
the combiistion chamber E, into which fuel is injected through feed line F and.
finally, after expanding, passes through the high-pressure turbine K and low-
pressure turbine H. Then the high- temperature gas exits through the exhaust
nozzle, which surrounds the outer ring nozzle with a cold current of air.
Figure 't?. Simplified Schematic Diagram of
the Operation of a Two-Circuit Jet Engine.
The air which has been speeded up through the fan of a turbofan engine
is exhausted with a slower speed than in the normal turbojet engine or the
normal two-circuit engine. The slower the speed of the flow behind the engine,
the lower the energy losses will be and the greater the engine's efficiency.
From jet- engine theory we know that the overall efficiency (overall Q-
f actor) for the power system of any aircraft is determined as the product of
the two basic figures: that of the internal (effective) and exhaust (flight)
The effective Q-factor increases with an increase in the air pressure in
the engine and with an increase in the gas temperature.
This leads to a substantial decrease in the specific fuel consumption.
Because only part of the air passes through the turbine in a two-system turbo-
jet engine, the turbine blades may be shorter than in a turbojet engine with
the same overall fuel consumption. For identical blade safety factors, this in /69
turn permits a 100 - 150° temperature increase in the gas in front of the
turbine, which gives a decided advantage over the turbojet engine in terms of
fuel economy. This is one of the reasons that the two-circuit and turbofan
engines have lower specific fuel consumptions.
For propulsive flight efficiency, from the theory of jet engines we are
familiar with the following formula:
where W is the speed of the Jetstream; and
V is the flight speed.
When the difference between the speed of the Jetstream and the flight
speed is decreased, i.e., when there is less of an uniised portion of the
kinetic energy, the propulsive efficiency increases and reaches its maximum
value (rij^ = 1) at a flight speed equal to the speed of the exhaust Jetstream.
When this is true, the unused portion of the kinetic energy is zero. A clear
example is the turboprop engine, in which the speed at which the air is thrust
back by the blade is close to the flight speed. However, in turboprop air-
craft the flight efficiency drops as the flight speed increases due to a drop
in the blade efficiency, and reaches low values at high subsonic speeds.
In two-circuit and turbofan engines, there is an increase in the area of
high efficiency, which the turboprop engine has at low flight speeds, up to
high subsonic speeds at which the flight efficiency is still too low.
To achieve this, in two-circuit and turbofan engines there is a second
circuit from which great masses of air flow at speeds close to the flight
speed, which aids in achieving a high flight efficiency as well as a low
specific fuel consumption. The specific fuel consumption for a two-circuit jet
engine and a turbofan engine is 0.52 = 0.65 kG fuel/kG thrust • hr for H =
and V = and 0.75 - 0.85 kG fuel/kG thrust • hr for H = 10-11 km at V = 750 -
In designing two-circuit engines, the selection of the two chief variables
is vital: the forward or rear positioning of the fan and the ratio of the mass
flow of cold air passing through circuit C to the mass flow of hot air passing
through circuit D, the so-called two-circuit level m = Gp/Gp^, whose value may
be from 0.23 to 3.5.
The two- circuit level is a vital engine parameter and determines its
efficiency, weight and thrust characteristics. The greater the level m, the /70
lower the specific fuel consumption; however, this entails an increase in the
engine dimensions and weight. At present the optimum degree is m = 0.6 - 0.7
for civilian aircraft at a flight Mach number of 0.8 - 0.9.
First-generation (Boeing-707-420, and Douglas DC-8) and second-generation
(Vickers VC-10 and others) transport aircraft are equipped with the Rolls
Royce Conway two-circuit engine in which m = 0.7 - 0.8. The engine thrust
for the Conway-509 is 10,200 kG, while the specific fuel consumption at top
conditions is 0.725 kG/kG • hr.
Even greater economy may be obtained through mixing flows of high
pressure (after the turbine) and low pressure C^^fter the fan) (in the JT8D
engine) or after the first ccanpressor stage (the Spey engine) in the exhaust
nozzle. When this is done, a relatively low speed of flow is achieved and
there is a correspondingly high efficiency. The combination of high thermo-
dynamic and thrust efficiencies has also made it possible to create engines
with low specific fuel consumptions. As an example. Table 7 presents some
data on the JT8D and Spey engines.
Tr. Note: Commas indicate decimal points
There are three JT8D engines on the Boeing- 727 and two on the DC-9, and
there are two Spey engines on the Bak- 1-11-200 and three on the Trident air-
craft. Soviet two- circuit engines were first installed on the Tu-124.
Replacing normal turbojet engines with two-circuit engines offers an
increase in payload and a decrease in the specific fuel consumption and the
As has already been stated, turbofan engines have the fans placed either
forward or behind. When the fan is placed behind, as was done by General
Electric (Figure 46d) , the design of the forward part of the engine differs
in no way from a normal turbojet engine: the compressor, the combustion
chamber and the gas turbine are identical. However, with turbofan engines,
after the gases have passed through the main turbine they run into one more,
the so-called fan turbine, which is mechanically tied in to the main turbine.
The blade tips in the fan turbine function as they would in a normal fan and,
in the annular gap between the nozzle and the additional turbine, they thrust
back a strong flow of air running parallel to the basic gas jet.
The American Convair 990A has four CJ-805-23B turbofan engines (built by
General Electric) with the rear fan, each generating a thrust of 7,300 kG.
The same engines are used on the French Caravelle-XA in replacement for the
obsolete Avon turbojet engines.
The Pratt and Whitney JT3D engine, with m = 1.5, has the fan positioned
forward. This type of engine is used on the Boeing-720B and DC-8. Table 8
offers some data on the JT3D engine.
Thus, use of two-circuit and fan engines makes it possible to create
aircraft with optimal flight characteristics for various purposes. The
increased thrust makes it possible to decrease the takeoff distance for any
specific aircraft weight or, in maintaining the takeoff distance, it becomes
possible to increase the payload or the fuel reserve.
Tr. Note: Commas indicate decimal points.
§ 2. Basic Characteristics of Turbojet Engines
In examining the flight conditions for turbojet passenger aircraft we
must know the following basic engine characteristics: thrust, specific thrust,
specific fuel consumption, specific weight and maximum-power altitude.
Thrust in turbojet engines is determined in accordance with the following
(W - V) kG,
where G is the per-second rate of airflow through the engine,
g = 9.81 m/sec^ is the acceleration;
W is the speed of the rate of gas flow from the exhaust
nozzle (m/sec) ;
V is the aircraft flight speed (m/sec) .
Turbojet engines designed in the last two decades have G
18 - 260
kG/sec, which corresponds to a thrust of from 800 - 900 to 10,000 - 13,000 kG,
W = 550 - 600 m/sec (stand-still operation), while in flight it reaches high
values. Two-circuit engines have a discharge velocity of 520 - 550 m/sec,
whereas turbofan engines have only 350 - 370 m/sec.
Specific thrust -- this is the thrust obtained from 1 kG of air passing
through the engine per-second:
Specific thrust characterizes the economy of an engine. In modern turbo-
jet engines, P
70 kG/kG/sec. Specific thrust depends strongly on
the compressor efficiency and turbine efficiency, as well as the degree to
which the air has been pre-heated. It determines the relative dimensions and
weight of the engine: the greater the specific thrust, the lower the engine
dimensions and weight for a given thrust.
Specific fuel consumption -- this is the relative hourly fuel consumed in
generating engine thrust :
c = p-^ kG • fuel/kG • thrust • hr,
where G is the hourly fuel consumption (kG fuel/hr) .
The specific consumption indicate how many kG of fuel have been expended
in creating 1 kG of thrust in an hour, and also characterizes the engine
efficiency. The lower the c , the more efficient the engine and the greater
the aircraft flight range and duration.
Specific weight of the engine is the ratio of the dry weight of the engine
to its thrust:
-^ kG/kG thrust.
In modem turbojet engines, g . = 0.19 - 0.35 kG/kG thrust. For example,
for the J-58 engine, the value of the specific weight is g . = 0.25 kG/kG
thrust. This means that for a thrust of 13,600 kG, the engine weight is
G . = 3,400 kG. As can be seen from these figures, turbojet engines do not
overload the aircraft by virtue of their weight. Whereas the weight of the
power system for a piston-engine aircraft may sometimes amount to 22 - 25% of
the takeoff weight, for turbojet aircraft this value equals only 10 - 12%.
§ 3. Throttle Characteristics
Depending on how it is used and on its rated service life, each engine
has several basic modes of operation which differ by the number of rpm's, the /73
temperature regimes, etc. Usually the following operation conditions are
distinguished: takeoff, nominal, cruising, and idling.
Practice in aircraft and engine use has resulted in the need for an
additional condition which, for the Tu-104 for example, has come to be called
the "extreme" condition. As can be seen from the very name itself, this is
used in only certain cases, specifically in the event of failure of one of
the engines. In this event, because of the engine forcing with respect to
the temperature of the supply of additional fuel and the increased revolutions,
the thrust increases by 8 to 10% by comparison to takeoff. However, this
emergency condition puts an overload on the engine which in turn means that
the engine must be overhauled faster than normally.
The takeoff condition corresponds to the maximum permissible number of
rpm's and the maximum thrust. Under this condition, the engine components are
subjected to the greatest mechanical and thermal stresses, as a result of which
their period of continuous use is limited and normally does not exceed 5-10
minutes. Takeoff conditions are applied to decrease the takeoff run through
increasing the horizontal flight speed, decreasing the aircraft acceleration
time and accelerating the breaking clouds in gaining altitude.
The normal rating corresponds to somewhat decreased (by 3-8%) rotation
with respect to the takeoff rating. The thrust is approximately 90% of the
takeoff thrust. The operation time at a normal rating is substantially longer:
it is used in gaining altitude and for near-ceiling flight. During such
operation the engine components are subjected to substantially lighter loads.
Cruising performance differs from the two preceding conditions through
decreased rpm's (by 10-15%) and thrust (by 25-50%) as opposed to maximum.
The idling period corresponds to the lowest number of rpm's at which the
eng'ine can operate stably. Under these conditions, there is little thrust
and therefore it is used in landing rims, dropping from high altitudes, etc.
The amount of thrust is 300-600 kG at low flight altitudes and 150-300 kG at
altitudes of 8,000 - 10,000 m.
shown in Figure 48
P ,, kli/kG'sec
of the change in engine thrust with respect to rpm's is
, from which we can see that an increase in the number of
Figure 48. Engin
and Specific Fuel
of the rpm's. n
e Thrust, Specific Thrust
Consumption as Functions
an increase in thrust.
At low rpm's, the amount of air
passing through the engine is
also low and as a consequence, the
fuel consumption, too, is low. The
amount of gases formed is small
and develop a negligible exhaust
velocity, so that the thrust
generated by the engines with this
value of rpm's is low, usually
300 - 600 kG. An increase in the
rpm's leads to a sharp increase
in the air exhaust, the fuel
delivery increases, the temperature
of gases in front of the turbine
increases and, as a result --
thrust increases. The highest
thrust may be obtained at the
maximum permissible rpm's, i.e.,
during takeoff or emergency con-
Figure 48 also shows the
dependence of the specific fuel
consumption on the number of rpm's
The change in cp is a function of
the degree o£ compression of the air in the combusion chamber. The more highly-
compressed the air is, the more fully the heat is used during the process of
fuel consumption and the lower the specific fuel consumption will be. Pre-
compression of the air depends basically on the compressor (engine rpm's) and
on the flight speed. Therefore, when the rpm's are increased, the specific
fuel consumption decreases. During normal and takeoff conditions, the specific
consumption is close to minimum.
Engine use during cruising rpm conditions yield optimum economy.
§ 4. High-Speed Characteristics
The high-speed characteristics of turbojet engines are the dependence of
the engine thrust, specific thrust and specific fuel consumption on flight
speed at a given altitude for a selected rule of control.
Let us examine the high-speed characteristics for constant rpm, gas
temperature in front of the turbine and flight altitude (Figures 49 and 50) .
Normally the characteristics are examined for a nominal number of rpm's.
From the formula P = —z — (W - V) we can see that the exhaust thrust will /75
be greater, the greater the amount of air which passes through the engine per
second and the greater the difference between the gas exhaust speed and the
flight speed. In increasing the flight speed from to 700 - 800 km/hr, thrust
decreases somewhat, because
G increases more slowly
than the difference W -V drops.
With an additional increase in
speed, on the other hand, the
increase in air exhaust begins
to surpass the decrease in the
differences between the speeds
W and V.
This is explained by the
character of the change in
thrust with respect to speed.
When the flight speed is
increased from to 700 - 800
km/hr, thrust decreases by no
more than 10-15%. This per-
mits us to consider the
,„^.T, ^ r-^- c available thrust generated by
l"^l"l I' /^?-"! A^^^ ' Function of ^ subsonic turbojet engine to
Mach Number (flight speed) for Various ^^ practically independent of
Altitudes (standard conditions, the broken flight sueed
line representing a temperature 10° above
T-0 = Take-off.
The specific thrust (?_„„£
W - V
) drops as the speed increases, because
the difference between speeds (W -V) decreases (Figure 50a) ,
The specific fuel consumption increases with an increase in flight speed
(Figure 50b) . When there is a change in the flight speed from zero to 750 -
850 km/hr, the specific fuel consumption increases by 15-30%. Thus, if for
V = the consumption is Cp = 0.89 kG/kG ' hr, then at a speed of 850 km/hr it
will increase to 1.15 (for the RD-3M engine). For the JT3D turbofan engine,
for V = 0, the consumption is Cp = 0.61, whereas for a speed of 880 km/hr it /76
is 0.781 kG/kG • hr (at an altitude of 11 km).
Figure 50. Change in Specific Fuel Consumption
(b) and Specific Thrust (a) with Respect to
Fl ight Speed.
In the supersonic speed range, the thrust in a turbojet engine at first
rises sharply, reaching its maximimii value at a specific speed, then sharply
decreases (Figure 51). Thus, for example, the
Figure 51. Representative Dependence of Engine
Thrust for a Supersonic Fighter on the Flight
Mach Number (H = 11,000 m) .
J- 79 engine (used in the
American F-104 fighter)
generates a thrust of 5,700
kG on the ground, which is
increased to 7,200 kG
through thrust augmentation
by afterburning. In flight
at altitude, the drop in
thrust is compensated by
velocity head. During
operation with afterburning
at an altitude of 11,000 m
and a speed of 2,000 km/hr
(M - 2) , it generates the
same thrust as on earth,
i.e. , 7,200 kG.
§ 5. High-Altitude Characteristics
The dependence of thrust, specific thrust and specific fuel consumption
on flight altitude for a constant number of engine rpm's and constant flight 111
speed is called the high-altitude characteristics.
The thrust of a turbojet engine decreases sharply with an increase in
flight altitude because thrust is directly proportional to the weight rate of
airflow, while the rate decreases with altitude due to a drop in air density.
The decrease in thrust with altitude occurs in spite of the fact that the
specific thrust, i.e., the thrust created by each kilogram of air passing
through the engine, increases by approximately half again as much as compared
to the ground level.
Up to an altitude of 11,000 meters, because of pre compress ion in the
compressor, the weight rate of airflow decreases more slowly than the air
density, whereas above 11,000 meters, where the temperature remains constant,
it drops more rapidly. The change in engine thrust with altitude may be
calculated with respect to the following formula: for altitudes up to 11,000
meters: P.. = P_ • aO-^; for altitudes greater than 11,000 meters: P., = 1.44
H U n
A • P- (here P„ is the thrust at altitude; P„ is the ground engine thrust);
A = — is the ratio of densities (A < 1") .
If we take P„ as 100%, then at an altitude of 10,000 meters the thrust
is approximately 45-50% of the ground thrust, while at an altitude of 20,000
meters it is only 10%. This comments on the lack of maximum-power altitude
in turbojet engines. However, modified turbojet engines developing a ground
thrust of 10,000 - 13,000 kG have high flight speeds at altitudes of 10,000 -
Figure 52 shows the variation in engine thrust in terms of altitude for
various rpm's. It should be noted that above the maximum-power altitude
boundary the power of piston engines drops more rapidly than does the thrust
of jet engines.
Up to an altitude of 11,000 meters the specific fuel consumption c
decreases, after which it holds constant (Figure 53). The basic principle in /78
the drop in c (and the increase in specific thrust) lies in the fact that
with a drop in the temperature of the surrounding air the degree of com-
pression in the compressor and the degree of pre compress ion are increased.
The hourly fuel consumption, which is equal to the product of c P,
decreases with an increase in flight altitude by approximately the same
intensity as does the air consumption and thriist.
The hourly fuel consumption at an altitude of 11,000 meters is less than
one half the ground consumption for the same engine rpm conditions.
Figure 52. Variation in Engine
Thrust in Terms of Flight Altitude
(Mach = 0.75).
Figure 53. Dependence of Specific
Fuel Consumption on Flight Altitude.
Thus, these engines are more effective in operation at high altitudes.
§ 6. The Effect of Air Temperature on Turbojet Engine Thrust
Air temperature, like altitude (pressure), has a significant effect on
thrust and specific fuel consumption.
During test-stand trial runs o£ the engine the measured thrust is
reduced to standard conditions, i.e., the so-called reduced thrust is deter-
mined for p = 760 mm Hg and t = 15 °C. Depending on the control system, the
effect of temperature changes on thrust is manifested in different ways. Thus,
for example, for turbojet engine with operational rpm's of 4,000 - 5,000, a
one-percent temperature increase decreases thrust by approximately 2%. For
two-circuit and turbofan engines with 6,700 - 11,000 rpm, a one-percent
temperature change varies the thrust by 1 - 1.5%. For example, the thrust
in a turbojet engine equals 7,000 kG for t = 15°C and p = 760 mm Hg. A
temperature increase of up to t = 25°C has occurred. Let us determine the
variation in engine thrust. To do so, let us express the temperature change
in a percentage ratio: T, = t°C + 273° = 15° + 273° = 288°; T =
= 298°; 298 : 288 = 1.03, i.e., the temperature increased by 3%.
thrust decreased by 6%, amounting to 420 kG.
25° + 273° =
Thus, for t = 25°C, the engine will generate around 6,600 kG of thrust.
If the temperature increases to 35°C, the thrust decreases by 13.6%, i.e.,
the engine will generate only about 6,000 kG of thrust.
When the air temperature increases, thrust increases. This comes about
because of the control system on the fuel-supply arrangement in turbojet
engines, which increases the fuel supply when temperature drops. An increase
in thrust usually occurs when the temperature decreases to +3 - -15°C,
depending on the engine conditions and the control o£ the fuel pump and
Let us determine the increase in thrust for a temperature of -15°C if
for t = 15°C thrust P = 7,000 kG: T^^ = 288°C, T^ = 258°C and 288 : 258 = 1.115,
i.e., the temperature increases by 11.5%, consequently, the thrust increases /79
by 23%, amounting to 1,600 kG (Figure 54).
Figure 5^. Effect of External Air
Temperature on Thrust of Turbojet
Eng i nes .
To maintain these engine
thrust values at high altitudes,
water injection into the compressor
Figure 55 shows the change in
thrxost in a JT3D turbofan engine
with and without water injection.
As can be seen from the figure,
water injection aids in maintaining
the calculated takeoff thrust up
to and intake temperature of +35 °C.
While this holds, the high-tempera-
ture flight characteristics for
the aircraft change negligibly. In
the case of the "Spey" engine, water
injection aids in forestalling a
drop in its thrust at temperatures
greater than 20°C.
§ 7. Thrust Horsepower
Thrust horsepower is the
available engine power:
Figure 55. Test-Stand Thrust in the JT3D
Turbofan Engine and the "Spey" - type Two-
Circuit Turbojet Engine as a Function of
the Ambient Air Temperature.
speed of 900 km/hr, if the available engine
where V is the flight speed in
Let us determine the thrust
horsepower for the engines of
an aircraft flying at an alti-
tude of 10,000 meters and a
thrust is 6,000 kG:
= 20000 hp.
However, at flight with the maximum speed of 1,000 km/hr at an altitude
of 6,000 m and with an available thrust of 9,000 kG, the thrust horsepower is
= 32000 hp.
The thrust horsepower increases directly proportionately to the speed.
When racing the engines on the ground without the aircraft's moving, N = 0,
because there is no work being done, i.e., PV = 0. A change in the available
horsepower with respect to altitude (rpm's being constant) is shown in Figure
Figure 56. Thrust Horsepower as
a Function of Mach Number for
Various Flight Altitudes (constant
rpm' s) .
In contrast to piston aircraft, in
which the available horsepower decreases
with an increase in speed above maximum
due to a drop in the propeller efficiency,
in jet aircraft it increases with an
increase in flight speed. Therefore,
rapid flight speeds may be obtained only
in aircraft with turbojet engines or
other types of jet engines.
Like thrust, the available horse-
power is a function of the engine rpm's:
the greater the number of engine rpm's
(for a specific altitude and flight
speed) , the higher the available horse-
§ 8. Positioning the Engines on the
The absence of propellers, the relatively low weight for high stress, and
their simplicity with respect to design and servicing make it possible to
install turbojet and turbofan engines in such a way that their optimal opera-
tional conditions and those of the aircraft are achieved.
At present, first- and second-generation turbojet passenger aircraft have
their engines mounted on the wing, on pylons below the wing, or in the tail
section of the fuselage.
Engine installation in wings. When the engines are installed in the wing
(between the upper and lower plankings), the total drag is reduced. In
practice, however, the engine is fastened to the fiiselage (in double-engine
aircraft), while the air duct extends along the chord in the wing. This leads
to a decrease in thrust as a result of a pressure loss in the duct, but in
contrast an advantage is the almost "clear" wing (without secondary structures)
which results. Engines arranged in this manner (close to the aircraft axis),
if one of them fails this creates only a slight turning moment.
Of the disadvantages which result from this arrangement, let us point
out the fact that it becomes impossible to make use of the thrust reversal
due to the heat effects of the gas jet on the fuselage (for a double- engine
aircraft) and the partial use of thrust reversal (for a four-engine arrangement)
(see Chapter IX). The stream of exhaust gases creates substantial noise in the
tail section of the fuselage and causes discomfort to the passengers seated in
the rear. On the Tu-104 and the Tu-124 (Figure 57), the engines are located
in the base of the wing, so that the greater part of the engine pod is hidden
behind the wing. In the De Havilland Comet, however, the engines are fully-
hidden in the wing (Figure 58). The engine's small size makes it possible to
design its pods with quite small maximum cross-sections.
Figure 57- The Tu-124.
Figure 58. The Comet
Engines located at the base of the wing create positive interference at
the most complex aerodynamic point -- the joint between the low-hung wing and
the fuselage. The effect of the Jetstream causes the formation of an "active
fairing" here, i.e., an increase in the "regeneration" of the surrounding flow.
This leads to a decrease in drag for the aircraft as a whole*.
However, this engine arrangement requires an increase in the relative
thickness of the airfoil profile, which causes a decrease in the aircraft's
Yeger, S.M. Design of Passenger Jet Aircraft (Proyektirovaniye passachir-
skikh reaktivnykh samelotov). Mashinostroyeniye. 1964.
high-speed characteristics. The angle at which the engines are installed
relative to the longitudinal axis is 3-5° in this arrangement. This inclina-
tion is necessary to guarantee that the engine exhaust flow does not hit the
elevator unit. In planform, the engines are turned outward by an angle of
2-4°, in order that the exhaust gas jet have less of an effect on the fuselage.
Positioning the engines on pylons beneath the wings. This is done on the
American Jets the Boeing- 707 and 720, the Douglas DC-8 (Figure 59), and the
Convair 880 and 990. Even the newly created Boeing- 737 shows a return to the
In this setup, the
positioning of the engines
increases aircraft drag
slightly, particularly due
to negative interference
from the wing and pylons.
However, the short length
of the engine's intake duct
when the air admission is
well designed minimizes
thrust losses and thereby
improve the aircraft's
Figure 59. Aircraft with Pylon Suspension
Suspending the engine
from a thin swept wing
substantially lightens the
wing and decreases its
structural weight. How-
ever, such a suspension requires increased reinforcement of the engine and
its pylon (due to greater inertial loads during aircraft maneuvering) and as
a result the wing weight is negligibly decreased. Aircraft with pylon sus-
pension of engines should be used only on concrete runways which have
substantially cleaner surfaces, because the engines are only 40-70 cm above
the ground. If foreign matter is drawn into the intake duct, the engine
compressor may fail. Although positioning the engines to the side of the
fuselage makes it possible to effectively use thrust reversal from all four
engines, the failure of the outside engine creates a substantial turning
moment, which greatly impedes handling the aircraft. This moment, acting in
the horizontal plane, causes an intense rolling motion around the longitudinal
axis, which (with allowance made for the aircraft's substantial moment of
inertia relative to the longitudinal axis) leads to an emergency situation.
The basic advantage of pylon engine suspension is the decreased noise
within the passengers' compartment.
Positioning of engines in the fuselage tail section. This arrangement
was first used in the French Caiavelle passenger aircraft (Figure 60). The
following aircraft have also been designed along these lines: the 11-62, the
Tu-134, the DC-9, the BAC-1 11, the Boeing-727, the De Havilland DH.121
Trident and the Vickers VC-10 (Figure 61).
Such an engine
arrangement yields the
"clear wing" and offers
maximum mechanization of
Jet passenger airlines
with such engine arrange-
ments have several ad-
vantages. The basic
advantage is their
characteristics and in-
creased comfort within the
passenger cabin (decreased
noise level) . The absence
of engine pods on the wing
results in negative inter-
ference being a factor only
at the juncture of the wing and fuselage. In addition, conditions are created
for designing a wing with an increased critical Mach number and a more
effective mechanical high-lift device on the wing. The lack of secondary
structures on the wing improves the wing's lift, which in turn permits a drop
in the wing area.
Figure 60. The Caravel le.
Figure 61. The Vickers
De Havi Hand DH.121 (b)
VC-10 (a) and the
Conditions are also created for the operation of the engine air scoops at /84
high angles of attack as a result of downwash, which in a sense "corrects" the
flow toward the side engine. During gusts, the entrance angle of the airflow
into the air scopp decreases almost to half the airfoil angle of attack, i.e., /85
when the airfoil angle of attack changes by 4°, for example, the direction of
the airflow around the air scoop varies by approximately 2°. The air will
enter the engine at less of an angle, which substantially decreases the pressure
loss at the intake. When the engine is installed in the wing or suspended from
a pylon, however, the entrance angle corresponds to the angle of attack at
which the aircraft is flying. Here the air circulation around the wing
increases the flow intake angle. As is well known, this causes additional
One of the structural characteristics of this arrangement is the T-shaped
tail assembly with its adjustable stabilizer. The elevator assembly, located
on the upper section of the vertical fin, is free from the destructive effect
of sound-waves created by the sound fields of the engine exhaust (Figure 62) .
This, too, has a specific effect in decreasing vibration.
Engine axis /s^^^__„ ^'I V
Figure 62. Diagram of the Effect of Engine Exhaust
Jets on the Stabilizer and Vertical Fin.
The aerodynamic advantage of the T-shaped tail assembly is that the flow
beyond the wing and its resultant separations have little effect on it during
The engine pods form horizontal surfaces which increase the aircraft's
longitudinal stability, in view of which the aircraft's longitudinal stability
characteristic progress linearly up to high angles of attack.
At the point of intersection of the horizontal tail surfaces and the
elevator for the T-shaped arrangement at high flight speeds, the increase in
drag drops as compared to the normal arrangement. This is an example of so-
called positive interference, and the effectiveness of the vertical tail
The engine pods have a horizontal pylon. The angle at which the pod is
set relative to the axis of the fuselage varies from zero to +2°, while in
the horizontal plane the pods may be turned out from the fuselage by an angle
of 2-4° (Figure 62).
Yeger, S.M. Design of Passenger Jet Aircraft (Proyektirovaniye passachir-
skikh reaktivnykh samelotov) . Mashinostroyeniye. 1964.
When the pod axis is higher than the structural axis of the fuselage and
consequently higher than the aircraft's center of gravity, a negative pitching
moment is created from the engine thrust.
Moving the engines to the tail section of the fuselage creates the /86
following operational advantages. As can be seen from Figure 63, only a slight
portion of the airflow thrust back by the nose wheels is covered by the engine.
The jets from the main wheels are covered by the wing both during takeoff and
landing. This decreases the possibility that foreign matter will enter the
engines off the runway. Ground maintenance of the engine is made simpler
through the ease with which the pods can be reached.
m^ r/nnnium /i nn ji mr !)
gg^— - jet
Figure 63. Diagram of the Effect of Airstream
Thrown Back from the Landing Gear Wheels: a -
engines mounted in wing; b - engines in tail
section of fuselage; c - engines on pylons.
When the engines are suspended from pylons, as was stated above, there is
no need for long air scoops. However, when the engines are mounted in the
wing, as was done in the Tu-104 and Tu-124 and the Comet, the length of the
air intake is 4-5 meters, as a result of which losses in air pressure at the
intake decrease engine thrust by 3-6%. Moving the engines to the tail,
however, decreases losses at the intake and the thrust drop is only 1-2%,
which improves the aircraft's takeoff performance.
In conclusion it should be noted that in spite of the numerous advantages
derived from installing the engines in the tail section of the fuselage, this
arrangement also has its drawbacks. Thus, for example, the engine performance
decreases at high angles of sideslip. The diving moment from engine thrust
increases both the speed of raising the landing gear nose wheels strut during
the takeoff run and the conditions for the control wheel. The need arises
for an adjustable stabilizer. There is an increase in the weight of the
rudder unit, which supports the elevator unit. The structure of the aircraft
becomes heavier as a result of the reinforcement for the construction of the
fuselage tail section due to the additional mass and inertial loads from the
engines as well as the need to increase reinforcement for the engines to /87
prevent its breakaway during emergency landing. During charging and fueling-
up, the aircraft center of gravity is shifted substantially farther forward,
which makes takeoff harder, and during flight requires precise functioning of
the automatic equipment which controls the fuel output.
Grouping the engines together in the tail section of the fuselage
facilitates using them for controlling the boundary layer (see Chapter IV)
and, finally, with the power plant arranged in this manner, the distance
from the engines to the ground is determined only by the aircraft's landing
configuration and the height of the landing gear. This makes it possible to
decrease the landing gear height and retain the permissible distance from
the ground to the edges of the air scoops.
§ 1 . Taxi ing
Aircraft with engines in the tail section of the fuselage or in the wing
(along the sides of the fuselage) have satisfactory taxiing properties. The
small thrust arm has no adverse effects on the aircraft's maneuvering pro-
perties. In fact, all modem jet aircraft have a pedal- control led leading
strut, which makes it easy to perform turns and maintain direction during
take off runs and landing runs .
The angle of rotation of the leading strut is 35-45°, while during take
off runs and landing runs (with flaps down) it is decreased to 5-6°. The
taxiing speed along the ground, during turns and close to obstacles reaches
no more than 10 km/hr, while in clear and straight runway sections, it is
no more than 50 km/hr.
Landing gears with nose wheels offer good runway stability during taxiing
on runways and taxiways . Turns are manipulated through the use of the leading
struts, as well as the creation of asymmetrical thinst and partial braking,
of the right or left landing gear trolley wheel. Turning an aircraft 180°
requires a runway 50-60 meters wide, depending on the width of the landing
gear wheels. Turbojet aircraft can also taxi over wet grass cover and over
unsmoothed snow cover at an airfield. The four to six wheels on each main
strut of the landing gear causes an even distribution of load over the air-
field surface, and reduced pressure in the pneumatic wheels (up to 4.5 - 6
kG/cm^) increases ability to travel over dirt airfields. Modem aircraft
using concrete landing strips maintain a tire pressure of 6.5 - 9.5 kG/cm^.
One drawback in the use of aircraft on dirt airfields is the damage to /88
the surface caused by the wheels during taxiing, takeoff and landing, the
formation of ruts, and the great amount- of dust thrown up from the exhaust
of the jet engines, which reduces visibility on the landing strip for pilots
of aircraft approaching for a landing.
§ 2. Stages of Takeoff
Takeoff is the aircraft's motion from the moment of starting until it
reaches an altitude of 10.7 meters* and has attained a safe flight speed.
This is the presently accepted altitude for complete takeoff according
to the ICAO and norms for flight worthiness for civil aircraft in the
The distance covered by the aircraft from the moment of starting iintil
the altitude of 10.7 meters has been reached is called the takeoff distance.
Aircraft takeoff (Figure 64) consists of two stages: a) taxiing until the
speed of lift-off and lift-off itself, b) acceleration from the lift-off speed
to a safe speed, with simultaneous climbing.
Figure 64. Diagram of Aircraft Tai<eoff and the Calculated
Takeoff Trajectory According to the iCAO: I - beginning of
run; 2 - takeoff run; 3 ~ acceleration and climbing; k -
point of aircraft lift-off; 5 - takeoff distance; 6 -
climbing trajectory for 100^ engine thrust; 7 ~ length of
calculated takeoff trajectory; 8 - permissible inclina-
tions in trajectory for extended takeoff due to engine
failure; 9 ~ actual trajectory of extended takeoff.
Immediately after lift-off, the aircraft's high thrust-weight ratio
permits it to gain altitude and accelerate up to its rate of climb along an
inclined trajectory. In this case, the gain in altitude is curvilinear,
because its angle of inclination constantly increases.
The holding after lift-off, which is used in the acceleration of piston
aircraft prior to beginning gaining altitude, is not applied in turbojet
The take-off run up to lift-off speed. As a rule, takeoff is performed
with flap deflection, from the brakes when the takeoff regime for the engines
is used. To this end, the engines are first put into takeoff rpm's and then
the brakes are slowly released. Figure 65 shows a graph of the coefficient
c as a function of the angle of attack and the aircraft polar for takeoff
position of the wing flaps and slats. An aircraft having triple-slotted flaps
(high value for c
) was used as an example.
Figure 65. The Dependence of c on a
and the Polars of an Aircraft having
Triple-Slotted Wing Flaps and Slats:
a - polar for aircraft with landing
gear down and wing flaps deflected at
25°; b - the same aircraft with
al lowance made for the effect of
screening by the earth during the
takeoff run (K = 1.6 : 0.134 = 12).
Note: T-0 = Take Off
arrangement of the flaps). For the example
At the beginning of the take- /90
off run, direction is maintained by
the brakes and directing the nose
wheel, and at a speed of 150-170
km/hr, when the rudder becomes
effective, it is maintained through
the appropriate inclination of the
rudder to the side as required.
When the proper takeoff angle of
attack (9-10°) is maintained, lift-
off of the aircraft from the
ground occurs without additional
movement of the control wheel when
lift-off speed is attained. With
a lift-off angle of attack of 9-10°,
the tail section of the fuselage
must be sufficiently far off the
runway and a specific sub-critical
angle of attack must be maintained.
If the pilot unintentionally
increases the angle of attack to
11-12°, contact of the tail
portion of the fuselage with the
concrete must be avoided.
An improperly chosen angle of
attack during lift-off may either
extend the length of the takeoff
run, or, on the contrary, lead to
premature lift-off at a low speed.
Thus, if the pilot achieves lift-
off at a lower angle of attack
(for example, with a = 6° instead
of 9-10°), i.e., below c ^ ,
which corresponds to a high speed,
the length of the takeoff run
increases. In calculating the
aircraft lift-off during takeoff,
the values normally accepted are
a = 8-11° and c , =1.3-1.7
(depending on the design and
shown in Figure 65, we have a, =
11° and c
Acceleration from the lift-off speed to a safe speed with simultaneous
climbing. Piloting an aircraft during this stage of flight involves the
following. After lift-off, maintaining the takeoff angle, the aircraft
smoothly shifts into gaining altitude with a subsequent decrease in the angle
o£ attack. The main wheels are braked, the time for complete braking averaging
0.2 - 0.3 sec. To decrease drag against the aircraft during climbing (after
lift-off), the landing gear must be retracted without delay. The aircraft's
hydraulic system retracts the landing gear, with opening and closing of the
main landing gear doors, in 5-15 sec. The landing gear is retracted at a
speed of 20-30 km/hr above the lift-off speed, and at a height not below
5-7 meters. During the process of retraction, the aircraft's speed increases.
After the landing gear is retracted, the flaps are in turn retracted at a
height not less than 50-80 meters, and the aircraft accelerates to a speed
for gaining altitude. The pilot must fly the aircraft during this interval
in such a way that before the flaps are retracted, the speed does not exceed
the permissible with respect to stability conditions. The time required for
retracting flaps deflected at a takeoff angle is 8-12 sec. As the flaps are
retracted, a pitching moment is created, so that pressing forces are created
on the control wheel which are easily relieved by the elevator trim tabs.
This is a case in which the electrical control of the elevator trim tabs is
convenient to use. After the flaps are retracted, the engine rpm's decrease
to normal and there is a further acceleration up to the climbing cruising
speed or to the flight speed along a rectangular root.
§ 3- Forces Acting on the Aircraft During the Takeoff Run and Takeoff /91
Let us examine the forces acting on the aircraft during the takeoff run
(Figure 66). The total force of the engine thrust acts in the direction of
the aircraft motion. The overall force of wheel friction against the ground
F = F^ + F„ and the aircraft drag Q act against the aircraft's motion,
braking it. The difference in the forces P -Q -F = R is called the
acceleration force. The following forces act perpendicular to the trajectory
of motion: lift force Y, force N of the reaction of the ground on the landing
gear wheels, and the force of weight G. The force R communicates to the
aircraft the acceleration
_^_acc _ ^cc .o. = q81^
where m is the aircraft mass .
The greater the acceleration force and the lower the aircraft weight,
the higher the acceleration will be. If instead of R we substitute its
value into the formula, we obtain
As the landing gear wheels roll along the ground, friction forces arise
whose value is a function of the condition of the runway (type of surface) and
IIIHII ■■■■III IBII
the degree of deformation in the tires. The amount of the force of friction
is detennined as the product of the loads on the wheels on the friction
a) Woment of friction force b)
Figure 66. Diagram of Forces Acting on the Aircraft
During Takeoff Run (a) and After Lift-Off During
CI imbi ng (b) .
During the takeoff run, the aircraft wing begins creating a lifting force
which rapidly increases and removes the load from the landing gear wheels.
The value of the friction force for each moment may be determined according
to the following formula: F = f (G - Y) . The friction coefficient (or
coefficient of adhesion) for dry concrete is f = 0.03 - 0.04, and for wet
concrete it is 0.05; for dry ground cover and for a cleared snow cover it is
0.07; for a wet grass surface it is 0.10.
The value P/G is the aircraft thrust-weight ratio during takeoff. The
greater the thrust-weight ratio, the greater the takeoff run acceleration and /91
the shorter the length of the takeoff run. Increasing the thrust-weight ratio
is an effective means of improving takeoff characteristics. For example, when
the Conway 550 double- circuit engines with their 7,500 kG thrust were
installed on the Boeing- 707, the thrust-weight ratio increased from 0.2 to 0.26.
A greater thrust-weight ratio is enjoyed by aircraft with two engines (0.28 -
0.33 kG thrust/kg weight), and the least is that of aircraft with four engines
(0.22 - 0.26 kG thrust/kg weight).
As can be seen from the formula above, the maximum acceleration is during
the first stage of the takeoff run (the aircraft drag force is low) .
With an increase in speed the thrust of jet engines decreases, although
during the takeoff run it may be considered constant. By comparison with
piston engines, the thrust of jet engines during takeoff decreases less
significantly and at the end of the takeoff run amounts to 87 - 92% of the
static thrust P^. The drag force during the takeoff run increases from to
Q,_ (aircraft drag at the instant of lift-off). At lift-off, Y = G, so
that the friction force will equal zero.
Thus, at the end of the takeoff portion, when the aircraft separates
from the ground, the acceleration force (reserve thrust) equals the difference
between the total engine thrust and the aircraft drag: R = P -Q.
Aircraft drag at the instant of lift-off (l-o) may be determined according
1-0 •'' 9 '
where c is the drag coefficient for an aircraft with landing gear down and
flaps extended in takeoff position at an angle of attack at the
instant of lift-off.
For example, for an aircraft with a takeoff weight of 76 tons and a wing
area of S = 180 m^ , the thrust during takeoff configuration for a lift-off
speed of 300 km/hr (83.3 m/sec) is approximately 17,000 kG. If we assume
that at lift-off c , = 0.07 - 0.075, then
X l-o '
Q.=c.. p5^ = 0.071.0.125 -180 -^=5500 kG.
'■ " 1-0 z 2
Then the acceleration force R = 17,000 -5,500 = 11,500 kG. The mean
ace ' '
acceleration at this instant will be
1 1 KflO
• = 1.48 m/sec
gRa.cc 9.81-11500 , ^o _ , 2
"■' a 76 000
The lower the value c ^ (due to the proper selection of the flap and
slat systems) , the lower Q^ will be and the greater the acceleration force
will be for the same assumed engiT^e thrust. For example, for an aircraft with
a low takeoff weight (two engines), during the takeoff run below the lift-off
speed R = 9,000 -5,800 kG, while the mean acceleration j = 2.5 - 2.0 m/sec2/93
In such an aircraft, the takeoff time decreases.
During the climbing portion of flight, under the effect of the force
(Figure 66) there will be a further increase i
case we may write the following equation of motion
R (Figure 66) there will be a further increase in flight speed. For this
R = P - Q - G sin 9 = mj ,
where G sin is the aircraft component weight acting along the line of
m is the aircraft mass .
Decreasing the total engine thrust with an increase in flight speed does
not decrease the value of the acceleration force, because as a result of a
decrease in the angle of attack, the induced drag for the aircraft decreases.
This allows an increase in the speed during the takeoff run portion (achieving
the required climbing speed or flight speed along a rectangular root) .
The length of the climbing portion with acceleration is a function of the
specific load, thrust-weight ratio, and other parameters.
The component G sin 9 initially has a low value, because the angle of
inclination of the trajectory during climbing is small (9=6- 10°; sin 9 =
= 0.105 - 0.175).
§ k. Length of Takeoff Run. Lift-off Speed
The length of the aircraft takeoff run is a function of the lift-off
speed and acceleration:
where i is the average acceleration value,
■^x ave "
The lift-off speed is determined according to formula:
/" or, , . . "1 /
- ^■,/_2^ = 14.4V — ^~ km/hr.
where k- is the unit load per 1 m of wing area.
The greatest unit load is in four-engined aircraft (the Super Vickers
VC-10, 570 kG/m2; DC-8-3C, 560 kG/m2) and somewhat lower in two-engined
aircraft (BAC-111-200, 370 kG/m2, the Caravelle-XB , 350 kG/m2) ; for three-
engined aircraft (the Boeing- 727 and the De Havilland Trident- IE) it is 450
For an average c , = 1.6 (triple-slot flaps and slats), the lift-off
speed for G/S = 450 - 500 kG/m2 is 220 - 240 km/hr. For an average acceleration
of j =2 m/sec2, the length of the takeoff- run is 1,100 - 1,300 m.
As has already been noted, the swept wing has a lower value for the /94
coefficient c then does the straight wing. This results in a lower value
y max " ^
for c , . All in all, this leads to a substantial increase in V^ , and
y 1-0 l-o
consequently in the length of the takeoff run. Therefore, the flaps and slats
are used to increase c . Deflecting them to their maximum angle at take-
y max ^
off may, of course, substantially decrease the lift-off speed, but in this
event there is also an increase in drag, a decrease in acceleration and, lastl)^
an increase in the length of the takeoff run. This requires selection of the
optimum angle of inclination for the flaps, at which c increases and.
consequently, so does c , , while the aircraft drag increases negligibly.
Designers are striving to achieve both the greatest value for c and high
aerodynamic performance in aircraft. If during takeoff the aircraft has a
fineness ratio of 14-15, this makes it possible to solve many problems such
as, for example, achieving the continuation of takeoff in the event of the
failure of an engine, decreasing noise in the area through a sharper climbing
trajectory, the selection of engines with optimal thrust values for a given
aircraft, etc. Calculations and flight tests have shown that the optimum
angle of deflection for flaps during takeoff is 10-25°. This angle yields
the optimum ratio between c , and c , which leads to a marked decrease in
^ y l-o x'
the length of the takeoff run. We must once more take note that c , is
^ y 1-0
selected from the condition of a sufficient reserve with respect to the angle
of attack prior to lift-off (c ), so as to eliminate sideslip. According
^ y max ^ ^
to norms of airworthiness, the aircraft lift-off speed must be no less than
20% greater than the brakeaway speed (see how it is determined in Chapter XI,
§ 5. Methods of Takeoff
Earlier we established that acceleration during the takeoff run and
consequently the length of the takeoff run are functions of the difference in
the available thrust and the overall aircraft drag. The engine thrust during
the takeoff run up to the lift-off speed of 220-240 km/hr varies insignifi-
cantly (by 6-8%). The overall aircraft drag during this portion of takeoff
is the sum of the aerodynamic drag (which increases as the angle of attack
increases) and the friction force of the wheels (on the runway surface), which
decreases as a result of a lessening of the load on the wheels then increase
in wing lift. Therefore, the pilot must select an angle a (different for each
aircraft) at which the total drag will be minimal and, consequently, the take-
off run will be shortest. Due to the lack of airflow of the slipstream from
the propellers, the effectiveness of the pitch control at the beginning of the
takeoff run is below that of a prop-driven aircraft. The required longitudi-
nal moment for lift-off of the nose wheel is created by the elevator only at
a rather high speed, close to the take-off speed. As a result of this, the
greater part of the take-off run for a turbojet aircraft is achieved in stand- /95
ing configuration. The angle of attack during the takeoff run is a function
of the angle (j) of the wing setting; if, for example, the setting angle (j) = 1°,
then a = 1° also. However, the wings of modem aircraft have geometric twist
(Chapter II, § 1), which creates an angle a which varies along this span. In
the graph shown in Figure 65, the value c corresponds to the average for
a takeoff run of a = 1 - 3°.
By the longitudinal position of the aircraft (the angle of the aircraft's
longitudinal axis), i.e., the angle of attack, the pilot may control in
achieving a speed at which the effectiveness of the elevator is sufficient
to initiate lifting the aircraft's nose (front landing gear strut). Often
this speed is selected from the condition of achieving rudder efficiency in
order to prevent the aircraft from turning on the main landing gear struts
with nose raised in the event of engine failure during the takeoff run. In
this event, the rudder should parry the turning moment from the asymmetric
thrust of the operating engines. Usually, after lift-off of the front strut,
the aircraft tends to progressively increase the pitch angle under the
effect of the increasing wing lift. Therefore, initially the control wheel is
brought back toward oneself, and then commensurably moved away, in an attempt
to maintain the aircraft at an angle of attack of 3-4°. The length of the
takeoff run is a function basically of the accurate setting of the angle of
attack. During the takeoff run, minor deviations from the optimum a, at
which drag is minimal, do not lead to a substantial increase in the length of
There are two ways of putting the aircraft into the takeoff angle of
attack. The first consists of the nose strut's lifting off at the instant
when elevator efficiency is achieved. The aircraft achieves an angle of
attack of 3-4° and the rest of the run takes place on the main landing gears.
Smoothly operating the elevator, the pilot maintains the angle of attack
during the takeoff run and at the instant of lift-off he creates the takeoff
angle of attack.
In the second way, which has only recently gained acceptance, the entire
takeoff run is performed in the standing configuration, and when a speed close
to the lift-off speed (V - 15 - 20 km/hr) is achieved, the control wheel
is smoothly but vigorously pulled toward oneself (in 4-5 sec), by which
motion the pilot lifts the front strut off and, without maintaining the air-
craft in a two-point configuration, puts it into the takeoff angle of attack.
Separation occurs practically from three points without any perceptible over-
load during the process of rotating the aircraft relative to the lateral
axis and increasing the pitching angle. In this way the pilot maintains
complete control of the takeoff run, the speed and the operation of the
engines. Usually during the takeoff run, the navigator states the aircraft
speed over the intercom at each 10 km/hr, starting at a speed of 150 km/hr,
while the pilot directs all his attention straight ahead. A controllable
leading strut simplifies maintaining the direction during the first stage of /96
the takeoff run, before the rudder becomes responsive, which almost eliminates
the use of the brakes in the main landing gear trolley.
In the second method of piloting, the takeoff distance remains practically
the same as in the first, but the takeoff run is somewhat shorter due to the
higher speed. Also, takeoff with a side wind is facilitated, since the
controllable nose wheel in combination with the rudder makes it possible to
hold a fixed direction up to the moment of separation without increasing the
takeoff run length (in aircraft with uncontrolled nose wheel, the run length
is usually increased due to the asymmetrical braking of main landing gear
trucks) , After the aircraft breaks away, the side wind causes it to turn
against the wind; for example, with a wind speed of 18-20 m/sec, the
rotation angle is 18-20°.
Flying investigations have shown that the required rotation of the
front wheel does not exceed 4-5° with a side wind up to 20 m/sec. This
allows the maximum permissible side wind during takeoff to be increased, for
example J a wind at 90° to the runway can be up to 15-18 m/sec, and also
simplifies the takeoff maneuver.
Up to the present time, no single opinion has developed among pilots as
to the way in which the control system of the front gear should be
constructed. The predominant opinion is that the rotation of the wheels
should be controlled by the rudder pedals (as on the TU-124 aircraft),
freeing the pilot's hands for operation of the elevator control lever, motor
throttles, etc. However, it is known that when the takeoff speed reaches
150-200 km/hr and the rudder begins to be effective, it is more expedient to
use the rudder alone to maintain the takeoff direction, disconnecting the
front landing gear, which is not always technically possible if the gear is
controlled by the pedals. Therefore, the wear rate of the rubber tires on
the front landing gear may be increased. A second plan is that of
independent control of rotation of the front landing gear, not connected to
the operation of the rudder (TU-104 aircraft) .
Let us analyze the technique of performing a takeoff using the second
method (separation from three points) . It is recommended that the elevator
trimmer lever be set at 0.5-0.8 divisions forward in advance, in order to
increase the load on the stick from the elevator at the moment of separ-
ation. Thus, these actions are in opposition to the established tradition,
according to which the trimmer control is moved 0.5-1 divisions back in
order to decrease loads at the moment of lifting of the front gear and
separation of the aircraft. Before beginning the takeoff run, the stick is
pushed forward approximately to the neutral position. Holding the aircraft
with the brakes, the engines are set at takeoff regime. After making sure
that the operating regime of the engines corresponds to the norm, the brakes
are released and the takeoff run is begun, during which the required
direction is maintained by controlling the front landing gear. The
effectiveness of control of the front landing gear is higher, the more
strongly the wheels are forced down to the runway. When sufficient effect- / 97
iveness of the rudder has been achieved to maintain the takeoff course,
generally 60-70% of the maximum speed, control of the front wheels can be
disconnected (if this is possible in the aircraft) . IVhen the takeoff is
being performed with a side wind, in order to prevent wind banking at the
moment of separation, the aileron control must be turned "against the wind"
by 30-80° with a wind speed of 8-18 m/sec before separation. After
separation, the rate of increase in the pitch angle must be slightly
decreased and the stick smoothly moved to the neutral position.
§6. Failure of Engine During Takeoff
Main takeoff characteristics of aircraft with one engine inoperative.
As we know, one of the main requirements placed on passenger aircraft is the
possibility of continuing takeoff and climb in case of engine failure. A
knowledge of the takeoff characteristics of an aircraft and timely usage of
the piloting recommendations in case of engine failure will guarantee a
successful continuation of the flight.
The takeoff characteristics of an aircraft with one inoperative engine
include the following: a) the length of the takeoff run from the starting
point to the moment of engine failure; b) the length of the takeoff run from
the moment of engine failure to the moment of separation; c) the inclination
of the trajectory during the climbing sector with acceleration; d) the
inclination of the trajectory during the climbing sector with landing gear
up; e) the critical engine failure speed (the speed of interruption of
takeoff") V ; f) the safe takeoff speed V ^ .
^ cr sto
If we know the length of the takeoff run of the aircraft from the
start position to the moment of engine failure and the length of the run
from the moment of failure to complete aircraft halt, which make up the
distance for interruption of takeoff, we can determine which airfields are
safe for operation of a given aircraft, which type of approaches to the
runway should be used, how the aircraft should be piloted with an inoper-
ative engine, etc.
In order to assure safety during continuation of the takeoff and climb
with one motor inoperative, it is necessary that the angle of inclination of
the takeoff trajectory and climb to altitude measured during tests be
greater than the minimum permissible angle (Figure 64) . As we can see from
the figure, after the landing gear are raised the inclination of the
trajectory should be no less than 2.5%, corresponding to an angle
e = 1° 30 min (sin 9 = V /V = 0.025 and = 1° 30 min) . The end of the
operation of raising the landing gear should correspond approximately to the
moment of passage of the takeoff distance (H = 10.7 m) plus 300 m.
In case of an engine failure during takeoff, the available thrust
decreases, the flying quality of the aircraft becomes worse and piloting
becomes more difficult due to the asymmetrical nature of the thrust and the /9j
low flight speeds, decrease in controllability and decrease in rate of
The decrease in available thrust leads to an increase in the dependence
of the flying characteristics of the aircraft on temperature and air
pressure. Therefore, the vertical speed of the aircraft with one engine
inoperative, characterizing- the possibility of continuing the takeoff and
climb under design conditions (p = 730 mm Hg and t = +30°C) are slightly
less than under standard conditions (p = 760 mm Hg and t = +15°C) .
The following speeds are characteristic for continued and interrupted
takeof fs : a) the critical speed of engine failure, V is the speed corre-
spending to the "critical point" during the takeoff run, at which failure of
one of the engines is possible. In case of failure of one engine at this
point, the pilot can either end the takeoff run within the distance
available, separate and continue his flight, or end his takeoff run and stop
within the interrupted takeoff distance; b) the safe takeoff speed
V ,is the speed at which the aircraft begins to climb after separation and
acceleration with one engine inoperative. According to the norms of the
ICAO, this should be 15-20% (depending on the number of engines on the
aircraft) greater than the separation speed for the takeoff configuration of
the aircraft: V^^^ >_ (1.15-1.2) Vg (see Chapter XI, §14).
If the speed of separation is less than the safe speed of the aircraft,
ircraft is held after
climb 'o altitude is begun.
the aircraft is held after separation with acceleration to V ^ , then the
The main characteristic indicating to the pilot that an engine has
failed is the appearance of a tendency of the aircraft to turn and bank
toward the engine which has failed. Also, failure of an engine can be
determined from the drop in oil pressure and fuel pressure, decrease in
engine rotating speed indicated by the tachometer, etc.
In order to make it possible for the pilot to decide to continue the
takeoff or interrupt the takeoff, the pilot should know the critical speed
for engine failure and for interruption of the takeoff.
During the process of aircraft testing, interrupted and continued
takeoff s are usually performed with one engine switched off during various
stages of the takeoff. When this is done, the length of the takeoff run to
separation of the aircraft and the length of the trajectory to altitude
10.7 m are measured if the takeoff is continued, as well as the length of
the run to halt if it is interrupted. When an interrupted takeoff is
performed, first the engine is turned off, then after 3 sec (reaction
of pilot to failure) the operating engines are reduced to the idle,
the spoilers are extended and the braking parachute is released and
intensive braking is begun. The transition to the idle is made due to the
necessity of maintaining pressure in the hydraulic system controlling the
spoilers and landing gear.
When a continued takeoff is performed, the pilot, after the engine is / 99
turned off, continues his acceleration to the separation speed and accel-
eration to the safe flying speed. The data produced by these tests are used
to construct graphs of the dependence of takeoff run, distance of continued
flight to H = 10.7 m and distance of interrupted takeoff on speed
(Figure 67) . The critical speed for engine failure (point B) corresponds to
point A of the intersection of the curves for interrupted and continued
takeof f s . Here also the so-called runway balance line in the direction of
the takeoff course (point C) is determined, which in case of an engine
failure during takeoff provides for continuation of the takeoff or stopping
of the aircraft (by braking) within the length of the runway after the
takeoff is interrupted.
up to Vj Ji tor .^Pi2-Zopt=-
on all engines | first engine out
— itor'Cont. Takeoff —
- iruy'^^l '-
Figure 67. Diagram for Determination of
Balance Runway Length and Critical Speed of Engine
Fa i 1 ure
If the takeoff is continued, acceleration of the aircraft to the safe
takeoff speed should be performed at an altitude of 5-7 m (above the
runway), at which point the landing gear should begin to be raised. At
10.7 m, the landing gear should be almost all the way up (takeoff distance).
The complete raising of the landing gear should be completed after the.
takeoff distance plus 300 m (reserve) have been covered.
In case of interruption of the takeoff, the run should be completed on
The critical speed for engine failure is the maximum speed, upon
reaching which the pilot can interrupt the takeoff or continue it with equal
safety. If the takeoff is continued at V < V (Figure 68), the continued
takeoff distance L^^^ to altitude 10.7 m is greater than the balanced
runway length; this is particularly dangerous if this length includes the
400-m terminal safety strip. This is a paved concrete strip (in case the
aircraft rolls beyond the actual runway during an interrupted takeoff) .
Figure 58. Vertical Speed of
Aircraft During Climb with One
Inoperative Engine As a Func-
tion of Flight Speed (Aircraft
= 35 t,
with Two Engines, G
Landing Gear Up
Figure 69. Vertical Speed
As a Function of Takeoff
Weight of Passenger Air-
craft (Aircraft with Two
Engines, Specific Loading
360 kg/m^, One Engine
Power 0.14 kg thrust/kg
In case of an interrupted takeoff at the separation speed V > V .
the braking distance will also be increased (point P) and the aircraft will
roll beyond the end of the airfield.
The best case is equality of critical speed and separation speed, since
this facilitates piloting of the aircraft considerably and makes it possible
to interrupt the takeoff safely right up to the moment of separation of the
Let us now analyze the selection of a safe speed for continuing of the
takeoff (Figure 68) . Usually at speeds of 280-320 km/hr, the maximum
vertical speed is achieved with the flaps in the takeoff position.
However, acceleration of the aircraft from V = 220-260 km/hr to a speed
of 280-320 km/hr requires a great deal of time and lengthens the takeoff
distance. Therefore, in order to avoid increasing the takeoff run length
unnecessarily, leaving it within limits of 600-800 m, the safe takeoff speed
is selected as 10-15 km/hr greater than the separation speed, if this will
provide a climb trajectory angle of no less than 2.5% for an aircraft with
landing gear up. With an average acceleration of 1 m/sec^, 3-4 sec are
required to increase the speed o£ the aircraft by 10-15 km/hr (2.8-
4.2 m/sec) . During this time, the aircraft can climb 5-7 m. The critical / lOl
speed of engine failure for an aircraft with a given weight under given
concrete atmospheric conditions for the balanced runway length has a
unique value. However, it is known that the engine thrust depends strongly
on temperature of the surrounding air and atmospheric pressure, and, for
example, decreases below the standard thrust with increasing temperature, so
that the excess available thrust decreases. This means that the takeoff run
length and takeoff distance increase, the vertical speed decreases
(Figure 69), the angle of inclination of the aircraft trajectory with a
continued takeoff with one engine inoperative decreases.
In order to go beyond the limitation with respect to trajectory inclin-
ation, the angle of inclination of the flaps must be decreased,' or if this
is insufficient, the takeoff weight must be decreased.
The operating instructions of every aircraft include graphs and
nomograms which can be used to determine the takeoff characteristics in case
of engine failure during the takeoff run. For this purpose, first of all on
the basis of the fact that the trajectory inclination of a continued takeoff
should be no less than 2.5%, the permissible takeoff weight is determined
for each selected flap angle and actual air temperature (Figure 70) . Then,
using the nomogram (Figure 71) for the same atmospheric conditions and the
weight which has been determined, the balanced runway length is found
(point K) . Then, using the nonogram (of Figure 72), the critical engine
failure speed (takeoff interruption) is found, as well as the safe speed for
continued takeoff. Figure 72 shows a nomogram for determination of the
critical speed. The same form of nomogram as on Figure 72 is constructed in
order to determine the safe speed for continued takeoff, takeoff run length,
separation speed, etc.
The nomograms on Figures 70-72 correspond to the norms of the ICAO.
The arrows on the nomograms show the path for determining desired quanti-
Piloting of an aircraft with one engine inoperative after separation.
Separation of an aircraft with one engine inoperative occurs at the same
speeds as with all engines operating. The effectiveness of the ailerons is
decreased. Therefore, the pilot should accelerate the aircraft to a safe
speed, exceeding the separation speed by 10-15 km/hr. This speed is also /102_
called the best takeoff speed, since it provides sufficient transverse
controllability and allows a climb to be performed at V :V = 2.5%.
Acceleration after separation should be performed near the ground,
since the aerodynamic influence of the surface is favorable and the
inductive drag of the aircraft is decreased. At V + 10-15 km/hr with
flaps deflected by 10-25°, a = 7-9° and the aerodynamic quality is 12-13;
the inductive drag (c = 1.15-1.3) is approximately equal to one-half of the
entire drag of the aircraft. With
quality values of 12-13, the thrust
consumption of the aircraft is
always considerably less than the
available thrust and the aircraft
can be either accelerated or trans-
ferred into a climb.
We can see from Figure 65 that
for an angle a
11°, the aero-
dynamic quality of the aircraft
K = 9, while considering the influ-
ence of the earth it is increased to
12. At 10-15 m, the influence of
the earth decreases sharply, and by
this time the aircraft is already
flying at the safe speed (in our
example this corresponds to a = 8°
and K = 9) . The airborne sector of
aircraft acceleration during which
it climbs to 5-7 m, is 600-800 m,
and the vertical speed V =1.5-
2.5 m/sec (depending on atmospheric
conditions) . Upon achieving the
safe altitude after acceleration,
the landing gear must be raised, in
order to decrease the drag.
6-8 sec after the landing
gear begin to come up, the drag of
the aircraft is decreased signif-
icantly and the excess thrust can
support a climb with higher vertical
speed, increasing the safety of
continuation of the flight.
Therefore, if the landing gear are
raised quickly, this should be done during the acceleration sector, although
the flying altitude will still be quite low. Raising the landing gear
increases the vertical speed by 0.5-1.0 m/sec, i.e., the climb will occur at
V = 2-2.7 m/sec (depending on the aircraft weight).
Climbing up to 100 m altitude should be continued at constant speed.
At this altitude, the aircraft can be accelerated to the permissible flight
speed with mechanical devices retracted, and the flaps can be raised. In
order to avoid a loss in altitude, it is recommended that the flaps be
raised in two to three partial movements. After the flaps are raised, the
engines should be set in the nominal regime. The direction of flight can be
maintained with one engine inoperative by deflection of the pedals and
creation of a 2-3-degree bank toward the engine still operating.
Figure 70. Nomogram for
Determination of Permissible
Takeoff Weight from Condition
of Production of Trajectory
Inclination of 2.5^ in Con-
10 20 30 to
Field Temp, °C
Takeoff wt. , T
Figure 71. Nomogram for Determination of Balanced
10 20 25 30 HO 50
Takeoff wt. , T
Figure 72. Nomogram for Determination of Crit
Engine Failure Speed
Flight trajectory with one engine inoperative. As we noted above, the
angle of inclination of the trajectory during the flight sector after the
landing gear are raised should be no less than 1° 30 min, i.e., 2.5%.
However, depending on the concrete conditions in which the aircraft is being
operated, this trajectory inclination may vary.
Under standard conditions, the aircraft has great vertical speed, so
that it is not difficult to provide the necessary trajectory angle. The
problem is somewhat more difficult under design conditions, and particularly
at high air temperatures, at which the vertical speed during takeoff with
one engine inoperative is sharply decreased.
Usually, the first marker beacon is located 900-1000 m from the runway,
and has a tower 10-12 m high. If the takeoff is continued, the aircraft
will fly over this point with landing gear almost up at fO-25 m. Errors in
piloting techniques and instrumental errors, as well as failure to follow
the flying instructions may result in reduced altitude of flight over this
beacon. It is therefore required that the approach to the runway be open in
order to avoid collision of aircraft with obstacles in case of a continued
§7. Influence of Various Factors on Takeoff Run Length
During the process of flying operations, the length of the takeoff run
may differ from the values calculated for standard conditions under the
influence of changes in engine thrust, aircraft weight, temperature,
density and pressure of the air, position of the flaps, speed and direction
of the wind.
Engine thrust has a clearly expressed dependence on engine rotation
speed. For example, if the rotating speed is decreased from the takeoff to
the nominal speed, the thrust is decreased by 5-7% (see Figure 52) .
Therefore, a decrease in rotating speed may increase the takeoff run length
considerably. During takeoff at the nominal regime, the takeoff run length
is increased by 10-12%, and flight safety in case of an engine failure is
The takeoff weight influences the takeoff run length as follows:
1) with an increase in weight, the separation speed increases; 2) with the
same engine thrust, an increase in weight leads to a decrease in perform-
ance, and consequently to a decrease in acceleration during the takeoff run.
As a result, the length of the run is increased.
The air temperature influences the takeoff run length in two direc-
tions. First of all, the air temperature influences the thrust of the
engine, and, secondly, it influences the true separation speed. Increasing
the temperature sauses a decrease in thrust, and consequently of
acceleration during the takeoff run, which increases the takeoff run length.
Also, increasing the temperature causes a decrease in density and,
consequently, an increase in the separation speed. For example, an increase /105
in air temperature of 10° increases the takeoff run length by 6-7%.
Pressure and density of the air. If the air temperature is constant,
but the pressure changes, the density of the air will also change; as the
pressure changes, the density changes by the same factor, since
p = 0.0473 £-,
where p is the air pressure, mm Hg;
T = 273 + t is the absolute temperature;
t is the temperature of the surrounding air in degrees Centigrade.
This formula allows us to determine the density in case of a simul-
taneous change of temperature and air pressure. A decrease in density leads
to an increase in the separation speeds and a decrease in the thrust of the
engine due to the decrease in the air flow by weight through the engine.
With decreasing thrust, the mean acceleration j decreases and, in the
final analysis, the takeoff run length increases. A decrease in pressure of
10 mm Hg leads to an increase in takeoff run length of 3-4%. Thus, during
takeoff under nonstandard conditions (t = +30°C and p = 730 mm Hg) the
takeoff run length is increased by 30-32%.
Wind speed and direction. The length of the takeoff run with a
wind is determined by the following formula:
where W is the head wind component of the wind (the "plus" sign is taken
with a tail wind, "minus" -- with a head wind).
The takeoff, as a rule, is performed against the wind, so that the run
length and takeoff distance are minimal. Separation occurs at a given air
speed V . With a head wind, the separation speed of the aircraft relative
to the ground is decreased by the value of the wind speed. Therefore, less
time is required for a takeoff run with a head wind than in calm air, and
the takeoff run length is decreased; while -with a tail wind it is increased.
For example, if the head wind speed is 5 m/sec [18 km/hr) , the aircraft need
be accelerated to only 222 km/hr ground speed, at which time the air speed
will be 240 km/hr, i.e., the separation speed is reached, and the takeoff
run is shortened. A headwind of 5 m/sec decreases the takeoff run length by
an average of 15-17%, while a tail wind of this same speed increases the
length by 18-20%. When taking off with a side wind, the aircraft tends to
turn into the wind, particularly during acceleration with the front landing
gear up. The reason for this rotation is the fact that aircraft with
turbojet engines have large vertical tail surface area, located at a
considerable distance from the main landing gear.
A quantitative estimate of the influence of various factors on the / 106
length of the takeoff run can be made using nomograms, with which the pilot
can determine the takeoff run length under the concrete takeoff conditions
§8. Methods of Improving Takeoff Characteristics
As we analyzed above, the length of the takeoff run depends on the
separation speed and acceleration during the takeoff rain. In turn, the
separation speed depends on the specific loading per 1 m^ of wing area and
c . while the acceleration depends on the excess thrust available,
A decrease in specific loading on the wing is the most effective method
of decreasing V and L^ . However, this always involves a decrease in
" sep tor
the useful weight carried, since with the surface area of the wing constant,
a decrease in takeoff weight can be achieved only by decreasing the useful
load. A decrease in the weight carried in a passenger aircraft means a
decrease in operational economy. Therefore, this means of decreasing the
takeoff run length is used to a limited extent, particularly since the
tendency to use the maximum possible flight range requires an increase in
specific loading on the wing.
The most acceptable method of decreasing the takeoff run length is an
increase in the lifting force of the wing using the wing mechanisms.
As we know, the main means of mechanization of the wing consists of the
flaps. All modern jet passenger aircraft have extendable (sliding) slit
type wing flaps^. The effectiveness of the flaps (magnitude of increase in
Ac ) increases as the slide (outward movement) of the flaps and angle of
flap deflection are increased. With low angles of flap deflection, the
lifting force is primarily increased without any essential increase in drag,
and the aerodynamic quality is decreased only insignificantly. These angles
can be used for takeoff during high temperature conditions, when the length
of the takeoff run can be retained within the required limits in spite of
the decrease in quality. The lower drag during the takeoff run allows a
considerable acceleration to be achieved.
Usually, attempts are made to produce the maximum aircraft aerodynamic
quality with the flaps deflected to the takeoff position, since the quality
determines the thrust consumed and the excess thrust which accelerates the
aircraft during the takeoff run. For aircraft with takeoff weights of
55-80 and aerodynamic quality of 12-14, a thrust of consumption of 5000-
6000 kg is required, and with a total available thrust of 13,000-28,000 kg,
the excess thrust provides rapid (25-30 sec) acceleration of the aircraft to / 107
the separation speed; the takeoff run length is 1000-1200 m.
Long experience of passenger aircraft operation has proven the useful-
ness of the method of decreasing takeoff run length by increasing the
available power (greater excess thrust) . The Boeing 727 aircraft carries a
^ S. M. Yeger, Pvoyektirovaniye Passazh-irskikh Reakti-vnykh Samotetov
[Design of Passenger Jet Aircraft], Mashi nostroyeniye Press, 196^.
three-slit flap (Figure 73) which, together with the slit type slat and
Kruger slat (front flap) makes it possible to produce c =2.7 with the
maximum angle of flap deflection. This in turn allows rather high values of
c to be achieved with lesser angles of deflection, corresponding to the
takeoff position of the flaps fc = 1.6-1.8).
^ y sep
Figure 73- Diagram of Extendable Flaps:
a, Single-slit (flow separation begins at 63 = 35"
^10°); b, c, Multi-slit (flow separation delayed
to 63 - 50-60°)
The multi-slit flap, due to the increase in curvature of the profile
and the pumping effect of the slits, delays flow separation to larger angles
of attack, which allows rather high values of c to be produced during
takeoff and landing. The increase in the lifting force of the wing with
flaps down results from a change in circulation around the wing with
increasing flow speed over the upper surface of the wing.
However, at large angles of attack, flow separation at the upper
surface begins at the front of the wing profile, which is combatted using
front slats or deflectable leading edges of the wing. Slit type slats
(Figure 74, a), which allow air to flow through the front slit, intensify
the boundary layer behind the peak of rarefaction on the wing profile and
increase the energy of the flow, so that separation of the flow is delayed
at high angles of attack.
When Kruger slats are opened (Figure 74, c) the effective aerodynamic
curvature of the provile is increased in the front portion, as a result of
which the load-bearing characteristics of the profile are improved. Since
this increases the suction force pulling forward, the drag of the wing with
the front slat open increases only slightly, and the aerodynamic quality
of the wing remains essentially unchanged.
The same effect can also be achieved by tilting the forward edge of the
wing downward (Figure 74, b) .
Thus, there is a rather large number of methods of increasing c and,
consequently, decreasing the separation speed and length of the aircraft
One promising method is the usage of the gas streams from the jet
engines. Experiments have shown that if the gas stream is directed down-
ward, it can supplement the lifting force of the wings. As a result, the
aircraft can be separated from the earth almost without a takeoff run.
During the landing, this same gas stream carries a portion of the flying
weight of the aircraft and allows the aircraft to be landed at low speeds.
Figure Ih. Diagram of Slit Type Front Slat (a),
Deflectable Front Portion of Aircraft Wing of
"Trident" Aircraft (b) and Kruger Front Slat (c)
The reaction flap (Figure 75) , a device consisting of a slit along the
rear edge of the wing through which a stream of air flows at a certain
angle g to the chord, driven by the compressor of the jet engine, is quite
important for heavy transport aircraft. This device changes the nature of
flow around the wing, causing a significant increase in lifting force. The
value of c increases due to the pumping of gas jets in the boundary layer
from the upper surface of the wing and the reaction of the outflowing gas
stream. The force of the reaction of the stream is divided into components
N and N . The component N increases the lift of the wing, while N
produces additional thrust. The lifting factor of a wing with a reactive
flap is equal to the sum of the lift factors of the aerodynamic effect of
the flow over the wing and from the reaction of the outflowing gases.
The usage of the reactive flap allows a broad range of flight speeds to
be used and simplifies the problem of takeoff and landing.
Systems are known for controlling the boundary layer, which either
remove or inject air. As we know, flow separation of the wing due to an
increased boundary layer thickness decreases coefficient c . By using
removal or injection in the boundary layer, the beginning of separation can
be delayed to higher angles of attack, which makes it possible to increase
the lift of the wing, decrease the takeoff and landing speed of the aircraft
and reduce the takeoff and landing run length (and consequently the length
of the runway). For example, a boundary layer blowing device decreases the
landing speed by 20-25%. This type of boundary layer control system (BLAC)
was used on the C-130C "Hercules" heavy turboprop transport. With this
system, the lifting force of the wing is increased more than when the
boundary layer is drawn off by suction. Four gas turbine reaction engines
located in two gondolas beneath the wing were used to supply compressed air
to the system. The air is collected in the rear portions of the gondola
and fed by four centrifugal compressors to a network of airlines (common
system for wing and tail surface) . Many small lines connect the main
distributing line with a common collecting chamber, from which the air is
ejected on the upper surfaces of the flaps and ailerons through slits. The
landing speed of the aircraft was decreased from 170 to 110 km/hr, while the
takeoff distance was reduced from 1280 to 853 m, and the landing distance
was reduced from 427 to 250 m.
ai r 1 ine
Figure 75- Reactive Flap on Wing (a) and Air
Feed System for Boundary Layer Injection at Wing
A BLAC system is also installed on the English Blackburn NA39
"Buckaneer" military turbojet aircraft. The experimental Boeing 707
aircraft used a system for boundary layer injection in the area of the flaps
using air taken from the engine compressors. During the tests, a decrease
in landing speed from 220-240 to 150-160 km/hr was achieved, i.e., by / HO
Turbofan engines expand the possibility for using BLAC in passenger jet
aircraft, since the removal of considerable masses of air from the outer
channel does not disrupt the operation of the engine.
The placement of a slat on the front edge of the wing and injection of
the boundary layer at the flaps and ailerons can produce a considerable
decrease in landing and takeoff speeds and allow the length of runways to be
decreased by 30-40%. The placement of a slat on the wing of a jet aircraft,
in addition to decreasing takeoff and landing speeds, also improves its
maneuverability at high speeds, since it delays the point of flow separation
to higher angles of attack. Practice has shown that slats can be used up to
M = 0.9.
A laminar flow control system is in the stage of development. It has
been experimentally established that the transition of laminar flow to
turbulent flow can be prevented by sucking the slow, turbulization-inclined
boundary layer away from the wing surface through a large number of thin
slots cut in the wing covering. This is called laminar flow control.
Investigations performed in the USA-' have shown that this method can-
increase the profile drag coefficient of a swept wing to a value near the
drag coefficient of a plate with laminar flow, i.e., decrease it by approx-
imately six times.
Laminar flow control by sucking away the boundary layer, naturally,
increases the load-carrying capacity of the wing. However, the usage of Ifc
to increase c alone is not expedient, since this problem can be more
simply solved by injection into the boundary layer. The production of high
aerodynamic quality (increased by a factor of 1.5 times) both during takeoff
and during flight, allows the takeoff and other characteristics of the
aircraft to be improved. Calculations have shown that for an aircraft like
the Lockheed C-141 with a takeoff weight of about 120 t and a cruising speed
of 850 km/hr, laminar flow control can increase the flight range by 30-33%.
With this flight range, the takeoff weight of the aircraft can be decreased
by 18-20% by decreasing the fuel reserves carried.
In conclusion for this chapter, we note that an improvement of takeoff
(as well as landing) characteristics of passenger jet aircraft -- decreased
takeoff run length and separation speed -- makes it possible to expand the
network of airfields and connect area and administrative centers. It is
always easier to find areas for small airfields than for large airfields. / Ill
Better takeoff and landing characteristics of aircraft will also provide a
lower "minimum weather" (see Chapter IX, §8) .
At the present time, considerable attention is being turned to the
creation of special passenger jet aircraft with short takeoff and landing
^ S. M. Yeger, Froyekti-vovani-ye Passazh-irskikh Reakti-vnykh Samotetov
[Design of Passenger Jet Aircraft], Mash inostroyeniye Press, 1964.
Chapter Vl . CI itnbing
Forces Acting on Aircraft
Climbing refers to straight and even (constant velocity) flight of an
aircraft in an ascending trajectory. During the climb, the forces acting on
the aircraft include the force of gravity G, the forcp of the thrust p',
lifting force Y and drag Q (Figure 76) .
Forces Y and Q are arbitrarily considered to be applied to the
center of gravity of the aircraft, although they are actually applied at the
center of pressure. This arbitrariness is permitted for forces Y and Q,
since the aircraft is balanced by deflection of the elevator. Force P for
simplicity of discussion will be considered to be applied through the
center of gravity. The direction of the effect of the forces is as follows:
force G acts vertically downward, force P -- forward at a certain angle 6
to the direction of flight, force Y -- perpendicular to the direction of
flight and force Q -- opposite to the direction of flight.
Figure 76. Diagram of Forces Acting on Aircraft in
Stable Climb: 1, Climb trajectory; 2, Longitudinal
axis of aircraft; 3, Chord of wing
The flight trajectory of the aircraft is inclined to the horizontal at
a certain angle ©, called the climbing angle. The following dependence
exists between the pitch angle &, the climbing angle 0, angle of attack a
and angle of wing setting (angle included between longitudinal axis of
aircraft and wing chord) : i? + cj) = © + a. For modern aircraft, angle
<j) = 1-3°, angle a = 2.5-5°, the pitch angle (the angle included between the
axis of the fuselage and the horizontal) in flight can be determined using
the gyrohorizon. During a climb, the climbing angle is less than the pitch
Force P does not correspond to the flight trajectory, forming with it a
certain angle 3. The magnitude of this angle is influenced by the angle of
motor setting relative to the longitudinal axis of the aircraft. As we
explained earlier (chapter 4, §8) the angle of motor setting may be from
zero to five degrees. Angle 6 can be determined as follows. Let us analyze
the climb during the first moments after takeoff. Let us assume that force
P forms an angle of 5° with the longitudinal axis of the aircraft,
the velocity in the climb is 520 km/hr, and the vertical speed is 16 m/sec.
The climbing angle can be determined as follows (Figure 76) :
i.e., == 6.5°. Then pitch angle i?=0+a-ct) = 6.5° + 3° - 1° = 8.5° (we
assume a = 3° for V = 520 km/hr, and the angle of wing setting cj) = 1°).
Since the difference between angles i? and for this case is 2°, force P
corresponds to the climbing trajectory, angle 6=7°. In this case, the
component P sin 6 is added to the lifting force. The magnitude of this
component may be rather high. For the quantities here being analyzed in an
aircraft with four motors with a thrust of each motor of 8,000 kg, we
produce P sin g = 32,000'0.122 = 3900 kg. This force is added to the lift
Y = 80-85 t.
As the altitude increases, the vertical speed decreases, but the true
velocity in the climb increases. Therefore, the lift angle is continually
decreased. We can therefore write the following two equations for a stable
y=G cos Q;
P=Q + G Sine.
We can see from the first equation that the lift during a climb equalizes
only a portion of the weight of the aircraft. The other portion of the
aircraft weight (G sin ©) is balanced by the motor thrust. For example,
for an aircraft weighing 38 t with a climbing angle 0=7°, component
G sin = 38,000-0.122 = 4630 kg, and for an aircraft weighing 80 t this
figure is 9770 kg.
If the available engine thrust for an aircraft with a takeoff weight of
38 t is 6700-7000 kg in the nominal operating mode (near the earth), more
than one half of this thrust is expended to balance the weight of the
aircraft, while the remaining thrust is expended in overcoming drag. The
climbing angle can also be determined from the second force equation:
G Q G K
where P - Q = AP is the excess thrust; P is the thrust factor of the aircraft;
the ratio of engine thrust to aircraft weight; Q/G is a quantity inverse to
At climbing angles of 6-8°, the value of cos ©
equation can be written as follows:
1, and the first
In order to determine angle &, we must use the Zhukovskiy curves for
consumed and available thrust. Figure 77 shows the definition of Ap , at
which the maximum climbing angle is achieved. The maximum excess thrust is
produced at the most favorable flight velocity, corresponding to the maximum
aerodynamic quality of the aircraft and the steepest climbing angle. For
aircraft with specific loads of 350-370 kg/m^, the most suitable speed is
360-370 km/hr, for specific loads of 500-550 kg/m^ -- 400-450 km/hr. The
excess thrust produced under these conditions at nominal engine operation
will provide a climbing angle © = 6-8°.
§2. Determination of Most Suitable
CI imbing Speed
The vertical speed in a climb is
determined by the formula V = V sin ©.
Replacing sin © with the excess thrust and
weight (we know from aerodynamics that
AP/G = sin 0, we produce
Figure 77- Determination
of Maximum Excess Thrust
Using Zhukovskiy Curves
In order to produce the maximum rate
of altitude increase (since it is this
quantity, not the climbing angle which is
of the greatest practical interest) , we must know the maximiom value of the
product APV, which represents the excess power: AN = APV.
For turbojet aircraft, the maximum values of the product APV kg*m/sec is
determined J and the vertical velocities are calculated (Figure 78) .
If we have the maximum values of the product APV/3.6(kg-m/sec) , we can
determine the maximum V for various weights .
The velocity along the trajectory at which the maximum rate of altitude
increase is achieved is called the climbing speed V
It is higher than the
speed at steepest climb which, as we showed in the preceding paragraph, corre-
sponds to the most suitable aircraft velocity (maximum quality) .
The climbing speed can be easily determined also using Zhukovskiy curves
for power consumed and available (Figure 79) (the available thrust power was
analyzed in Chapter IV ,§7, and the graph of power consumption for various
flight altitudes is constructed like the graph for thrust consumed) . In order
to do this, we must draw a tangent parallel to line N of power to the curve
for power consumed. At the point of contact, the excess AN
velocity corresponding to this excess power are determined.
300 hOO 500 600 "JOO f.XMJfxr
^hv ''cl ^^-^/^^
Figure 79- Zhukovskiy
Curves for Power
Figure 78. Excess Power
As a Function of Flight
Velocity (G^. = 52 T,
specific load 390 kg/m^)
For aircraft with wings swept at 30-35°, the maximum rate of altitude
increase is produced for practically all takeoff weights (from the maximum
permissible to the minimum with small commercial load) is produced at
indicated speeds of 480-550 km/hr at the earth. This speed must be maintained
up to 5000-6000 m. If this is done, the maximum rate of altitude increase
will be achieved at all altitudes. As the altitude increases, the true flight
speed will increase (for example at H = 6000 m and V
700 km/hr) .
Many flying investigations have shown that in order to retain maximum
vertical speed, the indicated speed must be decreased beginning at 6000-7000 m
by an average o£ 15-20 km/hr per 1000 m. Figure 78 shows that the product APV
has a smoothly sloping upper portion in the zone o£ maximum values, so that a
deviation o£ the indicated climbing speed of ±20 km/hr from the most favorable
value (pilot error) changes the vertical speed insignificantly, and the time
to climb and fuel expenditure over the climb remain practically unchanged from
the most favorable values.
The maximum vertical speeds of aircraft with two and three motors are
17-25 m/sec (at the earth), decreasing with increasing altitude to 8-10 m/sec
at 8000-9000 m. For aircraft with four motors, the vertical speeds are
12-15 m/sec at low altitude and 5-8 m/sec at high altitudes. The greatest
decrease in vertical speeds is observed at altitudes of over 10,000 m. The
flight altitude at which the vertical speeds equal 0.5 m/sec corresponds to
the practical ceiling of the aircraft. The height of the practical ceiling of
a passenger aircraft is 12,000-13,500 m. The height of the practical ceiling
(without consideration of maneuvering in the area of the airfield after
takeoff) can be reached by an aircraft in 43-45 min.
Figure 80. Vertica) Speed and Time of Climb for
An Aircraft witii Two Motors (nominal mode, power
factor P = 0.3)
Climbing at the nominal engine mode is the most economical (Figure 80),
since the maximum difference between available and consumed power is produced,
and the specific fuel consumption will be near minimal. A decrease in the
operating mode of the engines in a climb leads to an increase in specific fuel
consumption, a decrease in available power and rate of altitude increase of
the aircraft, an increase in climbing time, and as a result an increase in the
total fuel expenditure required to perform the climb. A modern passenger
aircraft reaches an altitude of 10,000-11,000 m in 18-25 min, covering
200-250 km and expending 2000-4000 kg of fuel (the higher values correspond to
three- and four-motor aircraft) .
§3. Velocity Regime of Climb
Climbing at the maximum rate of altitude increase is most economical. In
this case, up to 10,000-11,000 m the -climb occurs at an indicated speed of
460-440 km/hr (with corresponding lower true velocity), and upon reaching the
indicated altitude the pilot accelerates the aircraft at the nominal regime to
an indicated speed of 500-550 km/hr in 4-5 min for subsequent horizontal
flight at the maximum cruising regime. Thus, acceleration of the aircraft at
these altitudes, where the excess thrust is slight, requires additional time.
Operational tests of many turbojet passenger aircraft have shown that at times
it is more expedient (from the point of view of cost) to climb to altitude in
the so-called high speed regime.
To do this, the aircraft is turned in its final flight direction, then
accelerated to an indicated speed of 600-670 km/hr and the climb is performed
at this speed until the air speed reaches 800-880 km/hr (according to the thin
needle). At this point, the rate of altitude increase of the aircraft is .de-
creased to 12-14 m/sec, while the indicated speeds are considerably higher
than the most favorable speed.
When an air speed of 800-880 km/hr is reached, further climb is continued
at this speed. The rate of altitude increase decreases to 2-3 m/sec as
altitudes of 10,000-11,000 m are reached. The aircraft arrives at its
assigned altitude with sufficient true velocity, so that almost no additional
acceleration is required. After the transition to horizontal flight, the
cruising operating regime of the motors is instituted.
Climbing at the high speed regime decreases the duration of the flight,
but increases slightly the fuel expenditure. The problem is that as speeds of
600-880 km/hr are maintained, the vertical speed is decreased at all altitudes
and the time which the aircraft spends at low altitudes is increased, leading
to an increase in fuel expenditure in the climb. Therefore, the high speed
climb method is generally recommended for flights over short distances, 50-60%
of the maximum range of the aircraft with full fuel load. The additional /\\1
fuel expenditure in these flights requires no decrease in commercial load.
The distance which the aircraft travels in the horizontal direction
during the climb in the high speed regime is 50-100 km greater than during the
climb at maximum rate of altitude increase. The polar curve on Figure 81
characterizes these two climbing methods. As we can see from the figure, the
vector corresponding to the speed of 500 km/hr is directed more steeply upward,
corresponding to vertical speeds of 15-17 m/sec, while at 650 km/hr the
vertical speeds produced are less, but the horizontal range is greater.
Figure 81. Polar Curve of
CI imb ing Speeds
§4. Noise Reduction Methods
The noise of turbojet passenger
aircraft is caused by: oscillations of
cold air flowing around the aircraft
and mixing of the cold air with the
pulsating, hot gas jets from the
engines and oscillations of air com-
pressed in the compressors of the
The frequency spectrum of this
noise is significantly different from
the noise created by piston and turboprop motors. Whereas the noise spectrum
of turboprop engines is characterized by high sound pressures in the low
frequencies, the noise spectrum of turbojet engines contains predominantly
high frequency sound. This makes the noise created by a turbojet engine more
unpleasant to human hearing. The noise created by an ordinary turbojet at
over 35% thrust is greater than the noise resulting from the efflux of the
j ets .
The usage of two circuit turbojet motors allows the noise level to be
decreased during takeoff by 8-10 db (decibels), although the noise level is
still quite high. Existing engineering methods of noise reduction -- dampers
at the input pipes (JT8D engine) and exhaust nozzles (JT3D and Conway engines,
etc.) are not effective, and decrease the noise very slightly. For example, a
muffler on the output nozzle consisting of nine tubes decreases the noise
level by 5.5 db, but also decreases the efficiency of the engine. Install-
ation of perforated sheets and a screen around the air intake also provide
some decrease in noise level at the input to the compressor or fan.
Therefore, in order to decrease the noise to the required level (at high
power, the noise from the turbine and exhaust jet, at low power -- from the
compressor) , special methods of piloting after separation and during landing
must be used. As we know, foreign aircraft (the Being 707, Caravelle,
etc.) employ the so-called low noise takeoff and landing method (takeoff and
landing using the steepest trajectories with engines throttled over
listening check points), i.e., the decrease of noise at ground level is based
on rapid removal of the noise source from ground level. The initial climb is
achieved on steep trajectories at safe speed with decreased engine power.
This is aided by improved engine design and high mechanization of the wing.
In order to determine the influence of the noise of an aircraft taking
off on the population in the region of an airport, the quantity known as
perceived noise level is often used. It has been established that the
maximum permissible perceived noise level acting on the organs of hearing for
several seconds P
112 PN db (here PN db is the unit of measurement of
the noise). Noise levels over 112 PN db is said to be above the "tolerance
limit" for man.
At many large airports in Europe and the USA, limitations have been
placed on the noise created by aircraft taking off and landing. The apparatus
measuring the noise level is placed directly beneath the flight path of the
aircraft. If the maximum permissible noise level is exceeded, the airline
companies are forbidden to continue operating the aircraft.
Let us analyze the specifics of aircraft flight along a steep trajectory.
As we can see from the formula sin = V /V, in order to produce the maximum
angle 0, we must provide a combination of vertical speed and speed along
trajectory such that the value of sin is maximal. Flight tests are usually
performed to determine the steep climbing speed, during which the flaps are
left down at low speeds after takeoff in order to increase flight safety.
Therefore, the steep climbing speed is generally 40-50 km/hr higher than the
separation speed and practically corresponds to maximum aircraft aerodynamic
quality for the takeoff wing setting angle.
As is known, the flight regime with maximum trajectory inclination ©
corresponds to the maximum excess thrust AP and, consequently, the maximum
value of sin 0:
Therefore, if the most favorable aircraft speed (K ,0 ) is about
' ^ max max
350-360 km/hr for flaps up, due to the placement of the flaps in their landing /119
position, this speed is decreased to 300-310 km/hr. The climb after takeoff
on the steep trajectory is performed at the most favorable speed with flaps
During testing of one aircraft, the following method was developed for
steep climbing (Figure 82). With flaps down in the takeoff position (10°),
V = 260 km/hr. After separation, at an altitude of 5-10 m, the landing
gear was raised and the speed increased to 300 km/hr (at 50-60 m) . The
climb was continued to 300 m at this speed with the motor operating in the
takeoff mode, after which the motor was shifted to the nominal regime.
Whereas the climbing angle of the trajectory at the takeoff regime = 8-9°,
at the nominal regime it is decreased to 6.5-7°. At an altitude of 500 m, the
aircraft was decelerated by decreasing the vertical speed and the flaps were
raised. The flight was performed at a pitch angle of 14-16°.
During the landing, it is impossible to reduce noise by increasing the
steepness of the gliding t\-ajectory, since the rate of descent is fixed by the
operating conditions of the landing system. However, since the engines are
operating at reduced power, the initial noise level is decreased.
Figure 82. Optimal Climbing Trajectories for
Noise Reduction at Ground Level: a, Separation,
V = 260 km/hr; b, Beginning of lifting of
landing gear; c, Landing gear up; d, Accelera-
tion to V = 300 km/hr; e, Flight sector at
V = 300 km/hr; 63 = 10°; f, Beginning of accel-
eration for raising of flaps; g, Listening
point; h, Flight trajectory with continuous
acceleration; i. Point of beginning of lifting
flaps; j, End of lifting of flaps
The influence of noise from an aircraft taking off is particularly
noticeable if there is a populated point along the flight path at less than
4-5 km from the starting point of the aircraft. In such cases, tests must be
made to determine under which conditions and operating modes of the engines
permissible noise levels can be provided (in particular, 110-112 PN db for
takeoff during the day and 102 PN db at night, the "tolerance limit" for
noise being considerably lower at night). The nomogram on Figure 83 is
constructed from the results of flying tests on aircraft with two engines with
maximum takeoff weight under standard conditions of 38 T. The sloping lines
of the nomogram are the trajectories in steep climb situations.
The zero point on the nomogram corresponds to the beginning of the
aircraft takeoff run. On the right we have a table of operating regimes of
the engines and the corresponding noise levels perceived on the ground. The
dotted line shows an example of determination of the altitude of change in
engine operating regime and the necessary regime during takeoff of an
aircraft weighing 38 t when the edge of a populated point is located
3.3 km from the beginning of the takeoff run (the takeoff is performed during
the day, standard conditions, no wind). To do this, we draw a line from
point A, corresponding to a distance of 3.3 km, upward to the point of inter-
section with the 38 T weight line (point B) , then draw a horizontal line.
Point C determines the altitude (230-240 m) at which tie operating regime o£
the engines must be reduced to 88-89% (point D) , corresponding to the maximum
permissible noise level for daytime, 112 PN db. I£ the regime is not changed,
the noise level is 117 PN db (point D) .
After flying over the populated point or an increase in altitude of
500 m, the engines must be shifted to the nominal operating regime.
fingine speed, %
31 \ 93 \m 1
107 ■■ W1,5
1075 ■- iOZS
's 103,5-- 10H,S
mS -- 108 ■■ 103
Distance from start of rur} , KM
ni,5 \ 118
Figure 83. Nomogram for Determination of A)ti^
tude of Change in Operating Regime of Motor (con-
ditions of initial climb: V. , = 300 km/hr,
n= Sn, 63 = '0°)
As we can see from the same nomogram, with the same aircraft, but with a
separation distance to the populated point of 3.8 km (point E) , it is suffi-
cient to establish the nominal regime (point I) and maintain an altitude of
300 m (point F) in order to produce a noise level of 112 PN db in the daytime.
When the air temperature and pressure are changed or when there is a
wind, special graphs must be used to determine the corrected airc'raft weight,
since the flying data change. These graphs change for each aircraft in the
handbook on flying operations. For example, for the example above at
t = +25°C, p = 760 mm Hg with a head wind component of 2 m/sec, the corrected
weight G - 40 t with an actual weight of 38 t.
^ cor ^
The increased corrected
weight requires a lower altitude for the beginning of motor throttling.
However, the decreased operating regime of the engines after raising the
landing gear is not permitted at an altitude of less than 150 m.
In conclusion, we note that the flight speed during a steep climb to
altitude with flaps down should provide a sufficient reserve against
separation. The aircraft speeds at which horizontal flight with sufficient
controllability is possible is called the maneuvering speed; it must be
1.15 times the minimum speed corresponding to separation. For example, flying
tests indicate a minimum speed of 200 km/hr, so that the maneuvering speed is
230 km/hr. The reserve against separation with a steep climb speed of
300 km/hr is 70 km/hr, and the reserve to stall is about 100 km/hr.
§5- Climbing with One Motor Not Operating
If the situation requires a pilot to fly to a reserve airfield after a
motor failure on takeoff, with the reserve airfield located 350-400 km
distance, a climb must be performed. It will be shown in Chapter VII that
the most favorable altitude for ranges of 300-400 km is 5700-6000 m;
however, for flight with one motor not operating, the most favorable altitude
is 2500-3000 m. An aircraft with a motor out, when climbing at the nominal
regime, can attain a vertical velocity component of 3-6.5 m/sec at ground
level. This speed decreases with altitude and at 4500-7000 m, the rate of
altitude increase is about 0.5 ra/sec. It is considered that at this point the
aircraft reaches its practical flight ceiling with one motor not operating.
For aircraft with three motors, the flight altitude with one nonoperating
motor, naturally, is greater than for aircraft with two motors. The time to
climb to this altitude is 45-50 min and depends strongly on the actual
temperature of the surrounding air. The climbing speed in such cases is
70-100 km/hr less, explained by the decrease in available thrust of 30-50%,
so that the maximum of product APY is displaced toward lower values of
indicated (as well as true) speed. It is recommended that as the altitude is
increased, the indicated speed be decreased by S km/hr per 1000 m altitude.
Transition of engines from nominal to takeoff regime increases the excess
thrust and allows the rate of altitude increase of the aircraft to be
increased temporarily, although the time of operation in takeoff regime is
Chapter Vll. Horizontal Flight
§1. Diagram of Forces Acting on Aircraft
Horizontal flight means straight line, stable aircraft flight without
increase or decrease of altitude.
The forces acting on the aircraft were shown in chapter VI. We add that
the total aerodynamic force R (equalizing forces Y and Q) is applied at the
center of pressure, and is deflected from force Y by certain angle
(Figure 84) . Inclination of force R is changed by the pilot by using the
elevator, deflecting it enough so that force R passes through the center of
gravity. Therefore, we will consider for horizontal flight, as for climbing,
that all forces are applied to the center of gravity of the aircraft.
Figure 8^. Diagram of Forces Acting on Aircraft
in Horizontal Flight: 1, Longitudinal axis of
aircraft; 2, Chord line; 3, Direction of air-
craft; k, Direction of thrust
As we know, in order to achieve stable horizontal flight, it is necessary
that the following equation be fulfilled:
G = r+Psinp; Q = Pcosp.
These equalities show the conditions of horizontal flight. The first equality
shows that the movement of the aircraft is linear and occurs in the horizontal
plane. The second is the condition of evenness of motion, i.e., flight at
constant velocity. If this condition were not fulfilled, the flight would be
unstable (with acceleration or deceleration) .
It was stated above that force P may make a certain angle with the chord
of the wing. If we assume as an average a = 3°, the wing setting angle <J) = 1°
and the motor setting angle (in the tail portion of the fuselage) is zero, as
we see from Figure 84 angle 3=2°. Therefore, the force. P cos 3 will be less
than force P. In practice, with angle g = 2-7°, the value of cos 3 differs
little from unity, so that it can be considered that Q = P. We can also
consider that Y = G, since we can ignore the component P sin 3, which for
cruising thrust values will be less than one percent of the mean flying
weight. For example, with an average flying weight of 70 t and a quality of
14, the required thrust P = 5000 kg, and P sin 2° = 5000'0.035 = 175 kg,
i.e., 0.25% of the average weight. Even if ij) =5° (with engines in the rear
portion of the wing) and a = 3° and 3=7°, with the same P = 5000 kg we
produce P sin 7° = 5000-0.122 = 610 kg. This is 0.87% of the weight of 70 t.
§2. Required Thrust for Horizontal Flight
An aircraft is capable of performing flight at various angles of attack
within the speed range from the minimum to the maximum, i.e., at various
regimes. Each of these regimes corresponds to a certain air speed (angle of
attack), providing the lifting force equal to the weight of the aircraft.
This velocity has come to be called the required velocity for horizontal
flight, and the thrust necessary for the performance of horizontal flight at
this angle of attack is the required thrust for horizontal flight. Thus, in
horizontal flight a given angle of attack corresponds to a definite required
velocity and thrust. In order to calculate the graphs of required thrust on
Figure 85, a graph of the dependence c = f(a) and the polar curve of the
aircraft with a wing without geometric^twist is used. The calculation was
performed in the following order: the required thrust in horizontal flight is
set equal to the drag: P = Q. Setting various flight speeds, we determine
for each of them the impact pressure and c ; using the polar curve (for
various M numbers) we find the value of c corresponding to these speeds.
Using the formula P = Q = c Sp (V^/2) = c Sq, we determine the required
As we can see from Figure 85, with the most favorable angle of attack
a, = 6° and H = 0, we produce the minimum required thrust, corresponding to
the most favorable speed of 360 km/hr and quality K = 15 (from the formula
P = G/K we produce K = G/P = 35,000/2330 = 15). An increase or decrease in
speed leads to an increase in required thrust, since with angles of attack
greater than or less than 6°, the aerodynamic quality decreases. /124
For flight at 360 km/hr near the earth the motors must be throttled back
so as to achieve equality P = P . In this case, the curve of available
thrust touches the curve of required thrust at point B, corresponding to
a = 6°. As we can see from Figure 85, for flight with lower speed
(V = 300 km/hr) as well as for flight with higher speed (600 km/hr) ,
an increase in engine thrust is required (points C and A) .
300 ^^° WO
Figure 85. Required T'.irust As a Function of Flight
Speed (flying weight 35 T) : 1, Thrust for flight with
V, = 360 km/hr; 2, Thrust for flight with
l.g. = landing gear ^ = ^00 km/hr
We know that for aircraft with turbojet engines, the maximum excess
thrust corresponds to the most favorable speed and, in the example here
analyzed Vu , = 360 km/hr. In order to achieve AP at the takeoff or
^ nv max
nominal regime, an indicated flight speed of 360 km/hr must be maintained.
As the flying altitude is increased (for the same weight, in our example-
35 t) , the required thrust remains unchanged if the quality is the same. In
practice, however, as the indicated speed is retained, K decreases slightl)
with increasing altitude (by 0.4-0.6), so that P is somewhat higher. In our
example (Figure 85), the indicated speed of 360 km/hr at 10,000 m corresponds
to a true speed of 592 km/hr (M = 0.5) and a maximum quality of 14.5, i.e.,
the quality is decreased by 0.5. The angles of attack corresponding to K
are also different for different altitudes due to the influence of the
M number on the polar curve of the aircraft. For example, for H = 0, the
angle of attack corresponding to the minimum required thrust is 6°, and for
H = 10,000 m -- 4.8°.
A decrease in flying weight results in a decrease in required thrust for
the same angles of attack (and therefore, for the same altitudes). As we can
see on Figure 85, at H = 10,000 m for G = 30 t, the minimum P is less than
the minimum P for G = 35 t, and also the speed corresponding to the minimum
required thrust is less -- 575 km/hr (V. , = 350 km/hr) .
^3=25, i,g. down
Figure 86. Required Thrust As a Function of
Flight Speed (aircraft with three engines)
If we construct curves of required thrusts for aircraft with high weight
and specific load (for example with G = 80 t and G/S = 432 kg/m^) , the most
favorable speed is increased to 400 km/hr at H = and 625 km/hr at
H = 10,000 m (Figure 86).
In order to calculate the curves on Figure 86, we used the dependence
c = f(a) and the polar curve of the aircraft shown on Figures 16 and 27. The
increased a-^ is explained by the geometric twist of the wing, about 3°. For /126
clarity. Figure 86 shows the required thrust as a function of flight speed for
an aircraft with landing gear and flaps down, when the required thrust is
increased due to the decreased quality.
§3. Two Horizontal Flight Regimes
The points of intersection of the curves of required and available thrust
correspond to the equality P = P ; consequently, forces P and Q, as well as
Y and G will also be equal. On Figure 85 for H = 0, these points are marked
by the letters a, b and c. Due to the specific fdatures of piloting during
transition from one velocity to another, these points differ considerably.
For example, at point a the transition to a different speed requires simpler
control than at point c. Thus, in order to increase the speed to over
600 km/hr, acceleration must be performed by increasing the thrust (P > Q) .
In order to decrease the speed, the available thrust should be decreased,
since the required thrust in horizontal flight in this case is less than for
600 km/hr. However, in order to move to a different speed at point c, for
example, in order to increase thrust over 300 km/hr, the control stick must be
pushed forward to transfer the aircraft to a lower angle of attack and, in
order to maintain the same flight altitude, the thrust must be initially
decreased, then the necessary regime set when the speed begins to increase.
The same thing must be done to decrease the flight speed: the thrust must be
temporarily decreased, then once more increased, since a decrease in speed
causes an increase in required thrust.
Point a corresponds to the first flight regime, point c to the second.
The main peculiarity of the second regime is the necessity of double action
with the control lever of the motor when flight speed is changed. Therefore,
flight should not be performed in the second regime, since it decreases
controllability and makes flow separation on the aircraft wing possible.
The boundary between the first and second flight regimes is the most
favorable angle of attack for a turbojet aircraft (for a piston powered
aircraft it is the most economical). Whereas flights in the second regime had
no practical significance for piston powered craft, since flights at angles of
attack greater than the economical angle of attack were almost never performed
since a was near the maximum permissible angle of attack, flights of jet
aircraft (particularly at altitudes near the practical ceiling) may occur at
regimes near the most favorable.
The established minimum permissible operating speed on the basis of the
values c is usually 50-70 km/hr less than the most favorable speed. We
y per ^
should note that in the following in our analysis of examples we will not
consider altitude limitations related to the flying weight of the aircraft
(see §8 of this chapter).
In the examples on Figures 85 and 86, the division between the two flight /127
regimes at low altitude consists of the most favorable speeds of 360 km/hr and
400 km/hr. In horizontal flight with V, the motors must be throttled back so
that flight occurs at speeds corresponding to the point of contact of the
curves of available and required thrust (on Figure 85, point b) . As the
flying weight is decreased, the most favorable speed decreases; for example.
at 30 t, V „ = 350 km/hr indicated (Figure 85).
Lowering the landing gear and flaps displaces the boundary between first
and second regimes considerably toward lower speeds (Figure 86) . For example,
with flaps down the speed decreases to 325 km/hr (a - = 8.5°) and with flaps
down 25°, to 265 km/hr (ot -: = 7.8°). As a rule, the aircraft is brought in
for a landing in the first regime.
In order to avoid transferring to the second regime with the aircraft
wing mechanics in the takeoff and landing position, the pilot must recall the
indicated speed corresponding to the boundary between the two flight regimes.
%h. Influence of External Air Temperature on Required Thrust
As was noted, a change in the temperature of the surrounding air leads to
a change in engine thrust (chapter VI, §6). Also, temperature of the sur-
rounding air influences the nature of the dependence of required thrust on
flight speed, which appears as a displacement of the curve to the left (with
decreasing t) or to the right (with increasing t) and influences the value of
required speed for horizontal flight. The external air temperature does not
influence the required thrust, since P = G/K, and K = c /c depends only on
the angle of attack. Let us analyze the reason why the curve P = (V,t°) is
displaced. We know that in horizontal flight with unchanging angle of attack
(or c ) at different temperatures the following condition should be fulfilled:
As the temperature is decreased with constant pressure, the density of the air
is increased. In this case, in order for equality Y = G to be fulfilled, the
required horizontal flight speed must be decreased (c unchanged) . As the
velocities are decreased, the curves of required thrust will be shifted to the
left. As the temperature is increased, on the other hand, the curves of
required thrust are displaced to the right, since the required velocities
increase (Figure 87) .
As we can see from the figure, the same P corresponds to a greater
required thrust for a temperature 10° higher than the standard temperature, /128
since for t ^ we have V , , and for t ^ + 10° velocity V > V , .
St crl st ^ crl
The curves of required thrust for conditions other than standard are
calculated as follows. At first we find the air density under the new condi-
tions. For example, when the outside air temperature is increased by 10° with
pressure unchanged for H = 10,000 m, T = 223°K and p = 198 mm Hg, we produce
T = 223 + 10 = 233°, p = 0.0473 p/T = 0,0473-198/233 = 0.0403 kg-sec^/m'*.
This value of p, according to the standard table, is equivalent to a flight
altitude of 10,300 m.
Then, fixing the flight speed, we determine c , then take c from the
polar curve of the aircraft with various M (Figure 28) . Using the formula
P = c Sq, we determine the required thrust. In determining the M number, we
base our calculations on the fact that at T = 233°K, the speed of sound
a = 306 m/sec.
We must note that as the
temperature is increased by more
than 10°, the decrease in
density (increase in speed) will
be greater. For example, with
At = +30° at H = 10,000 m, the
decrease in density is
equivalent to an increase in
flying altitude to approximately
V km/ hr
Figure 87. Influence of Surrounding
Air Temperature on Required and
Available Aircraft Thrust (specific
loading 3^0 kg/m^)
speed, the available thrust will be equal
As the temperature is increased by 10° (by
required thrust is displaced to the right,
Let us now analyze the
graphs of required thrust
(Figure 87) .
With standard temperature,
in order to produce the velocity
10,000 m, we must
V , at H
use engine speed n.
to the required thrust (point A).
4.2% of 233°K), the curve of
and the curve of P is displaced
The available thrust, depending on the type and design of the motor, may
be decreased by 5-8% (curve 2) . The intersection of the curves of available
and required thrust defines the speed V „ with unchanged engine operating
regime. As we can see from the figure, the true flight speed has decreased,
so that the M number is also decreased, since the speed of sound is not 300,
but rather 306 m/sec (M = V 2/2O6) .
Thus, as the air temperature is increased by 10°, the flying regime
changes significantly. If we must maintain the previous M number (i.e.,
corresponding to t ), we must increase the operating speed of the engines
and, as we can see on Figure 87, set in engine speed n„% (point B) . The true
flight speed increases and becomes V _ = aM = 306 M.
^-?^:.*~-V,^\ '''-. .'
I If the pilot does not change the operating regime of the engines, as the
' flight speed is decreased from V to V _. the angle of attack and c
increase. Allowing the aircraft to fly at higher angles of attack is danger- ^
., ous due to the approach toward c and the separation limit. Also, under .'
' relatively high tenqierature conditions, the vertical gust reserve is
decreased. Therefore, in case such conditions are encountered, the rotating
speed of the engine should be increased by .an -average of 5% for each 5-10° ■:
of increase in temperature, or if this is impossible, a lower flying altitude
should be requested. :
As the temperature decreases, the available thrust increases (curve 4) [
] and the curve of required thrust is displaced to the left. The point of their t
1 intersection c defines the new flight speed. !
§5. Most Favorable Horizontal Flight Regimes, influence of Altitude and
The flight range is the distance traveled by the aircraft during the
climb, horizontal flight and descent. If flight is performed until the fuel
is completely exhausted, the distance traveled is called the technical range.
For passenger aircraft, the flight range given is usually that with one hour's
fuel reserve if the flight schedule is maintained, (recommended regimes).
Since there are various ways which the aircraft can leave the area of the
airfield and climb after takeoff, the range of flight covered during the climb
to assigned altitude changes significantly. However, the range covered during
climb to altitude is relatively. slight, so that in the following we will
discuss the range of horizontal flight.
The range of the horizontal flight sector depends on the fuel vesexve for
horizontal flight and on the rate at which it is expended, i.e., the kilometer
expenditure c, — the expenditure of fuel per kilometer of flight path.
Before going over to horizontal flight, the aircraft must take off and climb.
The fuel expenditure during the time of takeoff and climb to 9-11 km for two-
and three-engine aircraft is 1600-4000 kg.
The fuel expended during takeoff and establishment of nominal flight /150
regime (without consideration of climb) is 250-350 kg, the fuel expended
during the descent and landing is 700-1000 kg. In order to determine the
quantity of fuel to be used in the horizontal flight sector G _ , , we must
subtract from the quantity of fuel taken on board all supplementary expend-
itures and the navigational reserve. For example, with a takeoff weight of
the aircraft or 44,000 kg and an initial fuel weight of 13,000 kg, 7000-
7700 kg of fuel remain for horizontal flight at H = 10,000 m, since about
2000 kg are expended in takeoff and climbing, 800-1000 kg for descent and
landing and 2500 kg are held as navigational reserve.
For shorter range flights at the same altitude, the only change is in
the quantity of fuel required for the horizontal sector, while the remaining
fuel expenditure norms remain approximately unchanged.
The duration of horizontal flight is determined from the relationship
where c, is the hourly fuel expenditure.
The hourly fuel expenditure is the quantity of fuel expended by the
aircraft in one hour of horizontal flight. For example, for an aircraft with
three engines with a required thrust of 6000 kg and a specific expenditure of
0.8 kg/kg'hr, the hourly rate is 4800 kg/hr.
The relationship between hourly and kilometer expenditures is established
from the following considerations: in one hour of flight, the engines burn
c, kg of fuel. However, during this same time the aircraft covers a distance
numerically equal to the flight speed V (in calm air). Therefore, the fuel
expenditure per km is
where V is taken in km/hr. If V is taken in m/sec.
For V = 880 km/hr and c, = 4800 kg/hr, we produce c, = 5.46 kg/km.
Both the hourly and kilometer expenditures depend greatly on the
specific expenditure of the engines c . The relationship between the
specific and hourly expenditures is established as follows: for each
1 kg of thrust and one hour of engine operation, Cp kg of fuel are expended,
while a thrust of P kg requires the expenditure of P times more fuel.
.-■ W :i
In Chapter IV we established that the specific fuel expenditure depends
on the rotating speed of the engine, altitude and velocity of flight.
Let us now go oyer to an analysis of flight range. With identical fuel
reserve within the limits of possible speeds, various ranges will be produced.
For example, in the example outlined above with a fuel load of 13,000 kg, a
takeoff weight of 44,000 kg, flight at 10,000 m with a true speed of
810 km/hr (M = 0.75-0.76) and an hourly fuel expenditure of 2500 kg/hr, in
calm air a range on the order of 2800-3000 km can be produced. With flight at
a high M number (V > 810 km/hr), the range is decreased to 2200-2500 km.
Figure 88 shows- a flight profile for- an aircraft calculated for various
horizontal flight speeds, which also illustrates the above.
' 2800 L^M
Figure 88. Characteristic
Flight Profile of Aircraft
to Range at Fixed Altitude
A head wind or tail wind changes the
Let us analyze the influence of
flight speed on the hourly and kilometer
fuel expenditures. We can explain this
for flight at one and the same altitude,
using the Zhukovskiy curves for required
and available thrust (Figure 89) .
In order to achieve horizontal flight
at any given speed (V
it is required that P = P .
^ p r
^2 ^""^ V
that in order to fly at less than V ,
the engine must be throttled back so that the curve of P passes through
points A^ , A^ and A_ respectively (Figure 89 a) .
The hourly fuel expenditure c, = CpP , but since at any velocity of
horizontal flight P = P j
In order to decrease the flying speed, the rotating speed of the engine
must be decreased. This results in an increase in specific consximption.
However, as the flying speed is decreased, the value of P = G/K is also
decreased. Thus, as the engine is throttled back, Cp increases, but P
decreases. The hourly expenditure will depend on the way in which Cp and P
change. We find that as the flight speed is decreased, thrust P decreases
more intensively than c increases.
Therefore, c, also decreases: the minimum
c, . will correspond to V _, at which P . = G/K . With V < V -, c,
n mm ^ mf r mm max mf ' n
begins to increase, since P increases. Consequently, the greatest flight
duration at any altitude will occur when flying at the most favorable speed.
• ,kgi °}
"^ "' ''■ma^lhi ;■ ''mf '^ t, ^ . ^ , )f-/Kr
Figure 89. Explanation of influence of Flight
Speed on Hourly and Kilometer Fuel Expenditures
Let us explain how the flying altitude influences the hourly expenditure.
In §2 of this chapter we showed that the required thrust is almost identical
for the same weight at all flying altitudes up to 10,000 m. However, the
required speed increases with altitude. Therefore, the curves of required
thrust are displaced toward the area of higher speeds with increasing altitude
(see Figure 85) .
Since the available thrust of the engine decreases with altitude, the
curves of the change in thrust with velocity are displaced downward with an
increase in altitude. Therefore, whereas at low altitude the engines must be
throttled back, thus considerably increasing the specific expenditure, at
10,000 m less throttling is required and the specific expenditure increases
only slightly. When flying at the ceiling, the engines need not be throttled
back at all. Therefore, as the flying altitude increases the product c„P
-^ P r mm
decreases, which explains the decrease in hourly expenditure. Also, the
decrease in c, with altitude facilitates a decrease in specific expenditure at
constant operating speed. Therefore, the longest flight duration for an
aircraft with a turbojet engine is produced near the ceiling. Flight duration
at high altitude is 2-2.5 times greater than at low altitude. The regime
of lowest hourly expenditure is used when flying in a holding pattern or with
a strong tail wind (150-200 km/hr) in order to maintain the scheduled time of
Let us now analyze the way in which the selection of flight speed
influences the kilometer expenditure. It was shown above that c, = c,y
Substituting the value c, = c P in this formula, we produce
In Chapter IV we established that th
on the rotating speed of the engine, alt
Let us now go over to an analysis o
reserve within the limits o£ possible sp
For example, in the example outlined abo'
takeoff weight of 44,000 kg, flight at 1
810 km/hr (M = 0.75-0.76) and an hourly
calm air a range on the order of 2800-30
a high M number (V > 810 km/hr) , the ran
Figure 88 shows a flight profile for an
horizontal flight speeds, which also ill
Figure 88. Characteristic
Flight Profile of Aircraft
to Range at Fixed Altitude
the engine must be throttled back so tha
points A^ , A and A respectively (Figut
The hourly fuel expenditure c, = c,
horizontal flight P = P , c, = c_P . '
6 r p h P r
In order to decrease the flying sp^
must be decreased. This results in an j
However, as the flying speed is decrease
decreased. Thus, as the engine is throl
decreases. The hourly expenditure will'
change. We find that as the flight spe
more intensively than c increases. Th
If the pilot does not change the operating regime of the engines, as the
flight speed is decreased from V ^ to V „, the angle of attack and c
^ crl cr2 ^ y
increase. Allowing the aircraft to fly at higher angles of attack is danger-
ous due to the approach toward c and the separation limit. Also, under
relatively high temperature conditions, the vertical gust reserve is
decreased. Therefore, in case such conditions are encountered, the rotating
speed of the engine should be increased by an average of 5% for each 5-10°
of increase in temperature, or if this is impossible, a lower flying altitude
should be requested.
As the temperature decreases, the available thrust increases (curve 4)
and the curve of required thrust is displaced to the left. The point of their
intersection c defines the new flight speed.
§5. Most Favorable Horizontal Flight Regimes.
Influence of Altitude and
The flight range is the distance traveled by the aircraft during the
climb, horizontal flight and descent. If flight is performed until the fuel
is completely exhausted, the distance traveled is called the technical range.
For passenger aircraft, the flight range given is usually that with one hour's
fuel reserve if the flight schedule is maintained, (recommended regimes).
Since there are various ways which the aircraft can leave the area of the
airfield and climb after takeoff, the range of flight covered during the climb
to assigned altitude changes significantly. However, the range covered during
climb to altitude is relatively slight, so that in the following we will
discuss the range of horizontal flight.
The range of the horizontal flight sector depends on the fuel reserve for
horizontal flight and on the rate at which it is expended, i.e., the kilometer
expenditure c, -- the expenditure of fuel per kilometer of flight path.
Before going over to horizontal flight, the aircraft must take off and climb.
The fuel expenditure during the time of takeoff and climb to 9-11 km for two-
and three-engine aircraft is 1600-4000 kg.
The fuel expended during takeoff and establishment of nominal flight
regime (without consideration of climb) is 250-350 kg, the fuel expended
during the descent and landing is 700-1000 kg. In order to determine the
quantity of fuel to be used in the horizontal flight sector G^ , , we must
subtract from the quantity of fuel taken on board all supplementary expend-
itures and the navigational reserve. For example, with a takeoff weight of
the aircraft or 44,000 kg and an initial fuel weight of 13,000 kg, 7000-
7700 kg of fuel remain for horizontal flight at H = 10,000 m, since about
2000 kg are expended in takeoff and climbing, 800-1000 kg for descent and
landing and 2500 kg are held as navigational reserve.
For shorter range flights
the quantity of fuel required f
fuel expenditure norms remain a;
The duration of horizontal
c, . will correspond to V, _, at which P^ „• = G/K . With V < V ., c,
Ti mm ^ mf r mm max mr h
begins to increase, since P increases. Consequently, the greatest flight
duration at any altitude will occur when flying at the most favorable speed.
where c, is the hourly fuel expi
The hourly fuel expenditur(
aircraft in one hour of horizon"
three engines with a required tl
0.8 kg/kg-hr, the hourly rate i;
The relationship between he
from the following consideratioi
c, kg of fuel. However, during
numerically equal to the flight
expenditure per km is
where V is taken in km/hr. If \
For V = 880 km/hr and c, =
Both the hourly and kilomet
specific expenditure of the engi
specific and hourly expenditures
1 kg of thrust and one hour of e
while a thrust of P kg requires
*infj \m\v\ /'"/kkr
Figure 89. Explanation of Influence of Flight
Speed on Hourly and Kilometer Fuel Expenditures
Let us explain how the flying altitude influences the hourly expenditure.
In §2 of this chapter we showed that the required thrust is almost identical
for the same weight at all flying altitudes up to 10,000 m. However, the
required speed increases with altitude. Therefore, the curves of required
thrust are displaced toward the area of higher speeds with increasing altitude
(see Figure 85) .
Since the available thrust of the engine decreases with altitude, the
curves of the change in thrust with velocity are displaced downward with an
increase in altitude. Therefore, whereas at low altitude the engines must be
throttled back, thus considerably increasing the specific expenditure, at
10,000 m less throttling is required and the specific expenditure increases
only slightly. When flying at the ceiling, the engines need not be throttled
back at all. Therefore, as the flying altitude increases the product CpP ^^^
decreases, which explains the decrease in hourly expenditure. Also, the
decrease in c, with altitude facilitates a decrease in specific expenditure at
constant operating speed. Therefore, the longest flight duration for an
aircraft with a turbojet engine is produced near the ceiling. Flight duration
at high altitude is 2-2.5 times greater than at low altitude. The regime
of lowest hourly expenditure is used when flying in a holding pattern or with
a strong tail wind (150-200 km/hr) in order to maintain the scheduled time of
Let us now analyze the way in which the selection of flight speed
influences the kilometer expenditure. It was shown above that c^
Substituting the value c, = c P in this formula, we produce
In order to simplify our discussions, let us assume that C- remains
constant with changing flight speed, i.e., consider that neither a decrease in
engine thrust nor a decrease in the velocity itself influences c . Then it /t?'?
follows from the last expression for c, that the minimum kilometer expenditure
will occur at the speed for which the quantity P /V is minimal. In order to
determine this speed, we use the graph on Figure 89 b. The quantity
P /V = tan (j) (angle <p is formed by the horizontal axis and a ray from the
coordinate origin to any point on curve P ). When flying at V j-,
tan A = P . /V ., and when flying at V , tan d) = P /V
r mm mf ' ^ max' r max
We can see from the figure that with decreasing flight speed, angle (f>
decreases and reaches a minimum at a speed corresponding to the point of
contact of the ray to the curve of required thrust. This speed, at which P /V
is minimal, will be called speed V . With a further decrease in speed, angle
(|) begins to increase, i.e., P /V is increased. Thus, if we consider the
specific expenditure constant as the speed is changed, (P /V) . and conse-
quently also the minimal kilometer expenditure will be produced at speed V .
As we can see, V_ is always greater than V ^.
Let us now consider that the specific expenditure is not constant with
changing speed and consider the influence of throttling of the motor on
c„. If flight is performed at V , we have high P /V and nominal motor
P b f max ^ r
operating speed, so that c here is minimal. When we decrease the speed
(decrease motor operating speed) , we decrease P /V, but due to the throttling
of the motors, c„ increases. At V,, the value of P /V is minimal, but here
P 3 r
c is increased, since the engines are considerably throttled. Comparing
these two extreme positions, we might conclude that somewhere between V and
^ ' * max
V there should be a speed at which c P /V is minimal. This speed is slightly
greater than V and is called the speed of minimal kilometer expenditure. For
H = with a specific loading of 350-420 kg/m^, this speed is approximately
We can see from Figure 90 that as the altitude increases, the true speed
corresponding to the minimal kilometer expenditure also increases. We can see
from Figure 91 that the minimal kilometer expenditure decreases up to
10,800 m, then begins to increase. The decrease in kilometer expenditure of
fuel with increasing altitude is facilitated by the decrease in the quantity
P /V resulting from the increased flight speed and decreased specific fuel
In this example, the altitude of 10,800 m at which the minimum kilometer
expenditure is produced is called the most favorable altitude. For turbojet
aircraft it is 1000-1200 m below the practical ceiling, at which a consider-r
able wave drag is created due to the high angles of attack. Transition to
lower altitude, i.e., to lower angles of attack, decreases this drag component
significantly and increases the aerodynamic quality. Let us show that the
kilometer expenditure depends on quality:
Figure 90. Speed of Min-
imal Kilometer Expend-
i ture of Fuel As a
Function of Flying
Altitude (aircraft with
Figure 91. Influence of
F1 ight Altitude on Min-
imal Kilometer Fuel
Expend i ture
We can see from the formula that the kilometer expenditure is inversely
proportional to the quality. Now we can formulate a definition of most
favorable flight altitude: the altitude corresponding to (KV)
most favorable altitude or the altitude of least kilometer expenditure.
The dependence of the altitude of the practical ceiling and the altitude
of minimal kilometer expenditure on flying weight of a TU-124 aircraft is
shown on Figure 92, while Figure 93 shows the dependence of the minimal
kilometer expenditure for this aircraft on flight speed. We can see from this
last graph that the minimal kilometer expenditure is produced at
V = 752 km/hr. This is the speed V at the most favorable altitude.
Flights at lower and higher speeds and at other altitudes cause increases in
It has been established that at speeds 5-8% (30-50 km/hr) higher than
V , the kilometer expenditure is increased by an average of 1% (for
example, if c, . =3 kg/km, it will be increased to 3.03 kg/km) , and that
this is the optimal regime for long-distance flights. This cruising regime
is the most economical as concerns total transportation cost, since it
consumes little fuel, allowing higher commercial load to be carried.
For medium range flights (1300-1500 km), the highest cruising regime is
recommended, in which the kilometer expenditures are higher but the increased
fuel load does not require a decrease in commercial load, but the increase in
speed does decrease the flying time, as a result of which the cost of trans-
portation is decreased. These regimes correspond to flying altitudes of
7000-9000 m and maximal indicated speeds, or maximum permissible M number at
Ainijon on^ engirre
weight of aircraft
Figure 92. Height of
Practical Ceiling and
Height of Minimal Kilometer
Expenditure of Fuel As a
Function of Flying Weighit
700 75Z 800 V,KM/hr
Figure 93. Minimal Kilo-
meter Expenditure of Fuel
As a Function of Flight
Speed (aircraft with two
§6. Definition of Required Quantity of Fuel
In order to determine the fuel expenditure in flights to various
distances at various altitudes with various winds, a special graph must be
used (Figure 94). In calculating this graph, we assume the mean cruising
regime of engine operation, with a kilometer expenditure of one percent
greater than the minimal. This is sufficient to provide a fuel reserve
in case the flight is performed at higher or lower speed than the minimal
expenditure speed. The climbing and descending regimes for the aircraft
are identical in practically all cases. Therefore, the expenditures of
time and fuel for these pprtions of the flight can be considered constant,
dependent only on the flying altitude. The distance traveled by the aircraft
during the climb and descent also depends only on altitude.
When it is necessary to determine the flight range or fuel reserve
precisely under special conditions (special flights), a graph of this type
must be constructed for the regime selected. Figure 94 allows us to determine
without calculations the range of an aircraft for a given quantity of fuel
for any point. For example, point 4 corresponds to a fuel reserve of
7750 kg and a flight range (calm wind) of 2220 km at H = 10,000 m.
The lower portion of the graph presents corrections considering the
influence of wind.
. altitude ga^"
SOO woo 1500 2000 222DKH 2500 3000 3500
Distance between airports (S) ,
Figure 9^. Total Fuel Expenditure As a Function of
Distance, Altitude and Wind
If we must determine the fuel expenditure for flight of 1700 km at
8000 m with a tail wind of 175 km/hr, we move from point 1, corresponding to
S = 1700 km along the inclined lines for wind to point 2' corresponding to a
tail wind of 175 km/hr. Then we move vertically upward to the assigned
altitude of 8000 m (point 3') and here read the fuel expenditure: 5500 kg.
Adding the navigational reserve, we produce the quantity of fuel which must be
placed into the fuel tanks of the aircraft. For a flight of the same range
with a head wind of 80 km/hr (point 2) at 7000 m, 8000 kg will be required
(point 3) .
In processing the material of flying tests with respect to fuel reserves,
we usually determine the flying altitude most suitable as concerns total
flight cost. Table 9 presents these altitudes for one passenger aircraft.
As we can see from the table, even at 200-400 km range, the flight should
be performed at 4500-7000 m, since this will produce minimum fuel expenditure.
Flights over these ranges at 1200-1500 m (the altitude of the IL-14 aircraft)
are inefficient, since due to the comparatively low true flying speeds (570-
600 km/hr, indicated speed 480-550 km/hr) the kilometer expenditure is rather
Fl ight at the "Ceil ings'
With decreasing flying weight of the aircraft, the height of minimal
kilometer expenditure (most favorable altitude) increases (Figure 92) . This
dependence is used when flying at the "ceilings." The weight of the aircraft
when flying to maximum range can be reduced by 10-25 t (by 10-30% of initial
In order to keep the aircraft flying at all times at c, . , the
^ / 6 ]^ mm'
altitude must be gradually increased as the fuel is consumed. The density
should be decreased in proportion to the decreasing flying weight. This type
of flight is called flight at the ceilings. This is the way in which maximum
range can be attained. During the process of such a flight, the aircraft will
remain continuously at 1000-1200 m below its current practical ceiling.
We should note that civil aircraft perform flights at assigned altitudes.
However, it is of interest to the pilot to know the specific nature of flight
at the ceilings, since he may find this flight necessary, for example, when
flying along other than established airlanes and in other cases when maximum
range must be attained.
Let us analyze the performance of a flight at the ceilings (Figure 95)
using a TU- 124 aircraft. The initial altitude for this type of flight will be
10,500 m. This altitude (permissible on the basis of the condition of the
effect on the aircraft of a lO-m/sec vertical gust) will correspond to an
actual aircraft weight at the inning of the flight of 36 t (we will
consider that the flight is no- along an established airlane) .
At this altitude (p = 0,0395 kg-sec^/m'*, fuel weight 8400 kg), the pilot
should establish a horizontal flight speed of V , which in this case
corresponds to M = 0.7. This air speed will be maintained throughout the
entire flight. After approximately 2 hr 36 min, the pilot has expended
about 5200-5400 kg fuel, i.e., 15.5% of the initial weight. The air density
should be decreased by the same factor: 0.0395'84.5 = 0.0334 kg-sec^/m'^
(84.5% density at H = 10,500 m) , meaning that the aircraft will actually have
risen to an altitude of 11,800 m (see standard atmosphere table), i.e., will
have climbed by 1300 m, with a vertical velocity component of 1300/156 '60 =
= 0.139 m/sec. It is difficult to maintain this speed using the variometer,
piloting the aircraft by referring to the thin, needle of the KUS-1200 speed
indicator. In practice, it is easier to maintain the M number steady using
the M number indicator, since the value of a scale division of this instrioment
is 0.01. At 10,000-12,000 M, the air temperature, and consequently the speed
of sound, remains practically unchanged, so that with constant M number, the
true speed will also remain constant.
-*— — ■ ^L_ c-^
H0503-20O--ia30O kg t^^^=25mln+2 hr +^^=2^? kg
36 min+28niin = 3 hr 29 min
Figure 95. Profile of Flight at the
ceilings: a, At most favorable altitudes;
b, Ceiling; c, With altitude limited
according to flying weight
and increasing with increasing temperature)
In this example as
the weight is changed
for each 1000 kg the
flying altitude is
increased by 200-220 m.
For aircraft with
hourly fuel expend-
itures of 4000-5000 kg,
the increase in
altitude will be
50-70 ra. In flight at
the ceilings, the
rotating speed of the
engines and the M
number must be kept
constant. If the air
the engine rotating
speed should be changed
by one percent for each
5° (decreasing with
Flying tests have established that flight at the ceilings can increase
the range by 3-8%. Flight at the ceilings can be primarily used in case of
engine failure, when it is necessary to continue flying to the assigned
destination. It is here that the advantages of this type of flying are most
§8. Permissible Flying Altitudes. Influence of Aircraft Weight
The operation of jet aircraft with high practical ceilings (11,500-
13,000 m) has shown that it is not always possible to fly at these altitudes,
or even at the altitude of minimal kilometer expenditure (most favorable
altitude. Figure 92). The problem is that the flying altitude of a high
speed aircraft is selected on the basis of the condition of maintenance of a
reserve for overloads in case a vertical wind gust is encountered. Chapter XI,
will present an analysis of the effect of a vertical gust on an aircraft, and
now let us analyze the influence of aircraft weight on the selection of
permissible flight altitude, using the combined graphs c
Let us analyze the flight of a TU-124 weighing 34 t at 10,000 m at a
speed corresponding to M = 0.75, and explain the permissible overload in case
of a vertical maneuver from the standpoint of safety.
Q7 . 0,8h
Figure 96. Combined Graphs of
Dependences of Coefficients c
on M Number of Fl ight
As we can see from the
figure, for these altitudes and
M numbers the aircraft will have
c , - = 0.3 and c =0.715.
yhf y per
Consequently, the reserve with
respect to c will be
Ac = c ^- c , ^ = 0.715 -
y y per yhf
- 0.3 = 0.415. In case a
vertical gust is encountered or
in case of maneuver, this reserve
may be expended and the aircraft
will find itself at c
requires that the overload
C h.f .
value of c
This will be the value of
permissible overload. Each
M number (with unchanged
weight) corresponds to a definite
By joining the points corresponding to these values, we
produce the dependence c , ^ = f (M) (Figure 96) . As we can see from Figure 96,
in the range of numbers M
0.7-0.75, the reserve with respect to c is
With high M numbers, particularly at M > 0.8, the reserve of c is
decreased. This reserve is also decreased with increasing flight altitude
(with unchanged weight) and increasing aircraft weight (at constant altitude)
The reserve of c is equivalent to reserve against a vertical gust.
particular, it is required for a passenger aircraft that if an effective
indicator gust of 10 m/sec is encountered, the aircraft will reach only
c not encountering stall (see definition in Chapter XI) . Therefore;, in
order to avoid exceeding c and causing the aircraft to stall, permissible
flying altitudes are established as a function of flying weight (Figure 97).
If these limitations are not observed, a vertical gust of lower magnitude will
bring the aircraft to c or stall.
^ y per
The decrease in weight resulting from consumption of fuel increases the
reserve with respect to c and, therefore, the reserve for vertical gusts;
therefore, the flying altitude can be increased. In the same way as the
altitude is decreased (for example to 5000 m) , the reserve with respect to c
and gusts increases. For M = 0.6 (V
320*0.6 = 198 m/sec) , c
= 0.24 and c =0.92 (Figure 96). In this case, the overload permissible
with respect to c will be n = 0.92/0.24 = 3.83.
y y per
Figure 97 shows a graph of permissible flying altitude (for this
example) as a function of flying weight.
The standard practice of
assigning altitude intervals of
1000 m at altitudes above 6000 m
reduces the "resolving capacity" of
aircraft as to permissible altitude;
therefore, it would be more desirable
to use separations of 600 m altitude.
The heights of flight at the ceilings
correspond to permissible flying
35 a, tons
Figure 97- Permissible Flying
Altitude As a Function of Air-
craft Weight The limitation on flying
altitude is not the only limitation
for a high speed passenger aircraft.
The second limitation is the permissible M number for flights at high
altitudes (Chapter X-i §12). As flying operations have shown, the most
favorable cruising flight regimes as to M number and altitude for the first
generation of aircraft differ slightly from safe regimes as concerns the
conditions of encountering powerful ascending gusts .
§9. Engine Failure During Horizontal Flight
In case of engine failure, if^an aircraft cannot continue flying at
altitudes ordinarily used (8000-11,000 m) . As we know, in flights at alti-
tudes below the ceiling at speeds lower than the maximal, the engines are
throttled to some extent. This is also true of cruising flight regimes at
8000-11,000 m. The necessity of reducing engine speed in these regimes causes
an increase in the specific fuel expenditure. In case of failure of one
engine, the pilot will be forced to set the remaining engines at the nominal
regime (which is permitted for long term operation) , which should reduce the
specific expenditure. However, in this case the drag is increased due to
autorotation of the compressor and turbine of the engine which has failed
(for example, at V = 600-620 km/hr at 4000-5000 m altitude, the autorotation
drag is 150-300 kg) , leading to an increase in the kilometer and hourly
expenditures. In case of an engine failure, horizontal flight at altitudes
above 6000-7000 m is impossible, and the aircraft will descend to 5500-6000 m
(two-engine aircraft. Figure 98). For aircraft with three and four engines in
case of failure of one engine, the decrease in altitude is not so great.
The altitude at which
the aircraft can fly
without further descent
will be essentially the
initial altitude of flight
at the ceilings with one
nonoperating motor, if
long range flight must be
performed and a landing
cannot be made immediately
after the motor fails.
Figure 98. Profile of Flight of Aircraft-
with Two Engines in Case of Failure of One
Engine After h^ min Flying Time: a, Point
of failure; b, Descending trajectory (time
37 min, L = 400 km); c, Flight with
In case of a motor
failure, it is necessary
first of all to achieve
the least possible rate of
vertical descent and
secondly to decrease the
weight of the aircraft
rapidly (using up fuel) in order to make it possible to continue horizontal
flight with one nonoperating engine at high altitude. Therefore, the descent
should be made at the nominal regime, gradually decreasing the vertical
velocity component, which at the beginning of the descent will be
V = 3-5.5 m/sec. The indicated speed for each aircraft depends on the
specific loading on the wing and the power factor. For example, for an
aircraft with two engines and a specific loading of 350 kg/m^, an indicated
speed of 430 km/hr was produced. The descent from 10,000-11,000 m to the
practical ceiling of the aircraft with one nonoperating engine occurs in
35-45 min. Over this time, the aircraft covers 350-500 km.
If it is necessary to continue the flight, the pilot should shift the
aircraft to the regime of flying at the ceilings; then in 60-70 min the
aircraft will cover another 650-750 km, with an increase in altitude of
800-1000 m and an average rate of altitude increase of 0.15-0.2 m/sec. Flight
should be performed at M = 0.50-0.55, corresponding at 5500-6500 m altitude to
a true speed of 600-650 km/hr. The mean kilometer fuel expenditure for an
aircraft with two engines at this stage will be about 3.5 kg/km, which is
approximately 0.5 kg/km greater than at 10,000 m with two engines operating.
Thus, the flight range with one engine not operating is always less.
A gain in flying range with one engine not operating can be produced only
if the initial flying weight was planned (due to unavailability of higher
altitudes or other reasons) for a low altitude, for example 6000-7000 m. For
example, for the TU-104 aircraft at this altitude at 800 km/hr, the hourly
fuel expenditure is 3100 kg/hr, and the kilometer expenditure is 3100/800 =
= 3.88 kg/km. In case one engine fails, it is possible to fly at 5000 m and
620 km/hr, the second engine operating at the nominal regime with an hourly
expenditure of 2200-2300 kg/hr. In this case the kilometer expenditure will
be about 3.6 kg/km, i.e., less than in flight with both engines (for this
altitude) and the possible flying range increases.
In all cases in case of failure of one engine, the crew should return
to the airfield of origin if possible or land at the nearest available
§10. Minimum Permissible Horizontal Flight Speed
The most favorable horizontal flight speed is the division between the
two flight regimes. However, in establishing the minimum permissible speed,
the most favorable speed is not taken into consideration, but calculations
are based on c , produced "for low M numbers. The value of c , which
y per ^ y max
is used to determine the stall speed, is also not used in this case.
Let us determine the minimum speed of horizontal flight, i.e., the speed
corresponding to c , assuming that the wing area is 120 m^, the aircraft
weight is 50 t, and c =1.2 (from the graph on Figure 96):
When values of c > c are achieved, the stability of an aircraft /143
y y per ' ^
with a smooth wing (flaps up) may be disrupted. In order to prevent a loss of
speed and a stall, the minimum permissible horizontal flight speed should be
50-60 km/hr greater than the absolutely minimal speed. In our example, this
will be 320 km/hr. After 10 t of fuel have been expended (G. ^ = 40 t) we
produce (according to the last formula) the minimal possible speed of
240 km/hr, so that the minimal permissible speed will be 300 km/hr.
Frequently, in order to avoid the necessity of memorizing many values of
minimal permissible speed, flying handbooks show only the value for maximum
weight. In our example, this will be 320 km/hr. When flying at this speed,
an aircraft weighing 40-50 t or less will have c < c by 30-40%. With
^ " y y per
normal operation of the aircraft, flying at 320 km/hr is not permissible,
since even for circle flights the speed at this weight (S = 120 m^) should be
This limitation will provide flight safety.
Chapter V I I I . Descent /I43
§1. General Statements. Forces Acting on Aircraft During Descent
Descent refers to steady, straight line flight of the aircraft on a
descending trajectory. Descent at low power, when the thrust at 8000-
10,000 m is flight, will be called gliding. Usually, passenger aircraft
descend with the engines operating at 80-86% revolutions, at which the thrust
is greater than at the idle (for example, the idle at H = 10,000 m might
correspond to 72-74% revolution) . The presence of motor thrust increases the
descent range and decreases the angle of inclination of the trajectory.
Following his assigned altitude (9000-11,000 m) the pilot begins his
descent at 250-300 km from the airfield at a high indicated speed
(550-650 kra/hr) . The time for the beginning of the descent is calculated by
In those cases when the flight range is not over 1000-1200 km and fuel
economy is of less significance than flying time economy, the descent is
performed at the greatest permissible indicated speed or M number.
Figure 99 shows the forces acting on an aircraft during the descent with
engines operating. The angle of inclination of the trajectory of the descent
from 9000-11,000 m will be = 2.5-3°, the pitch angle ^ ^ 2-2.5°. It must be /144
be noted that angle does not remain constant, but rather changes as a
function of the vertical component of the descent, which is maintained by the
pilot by setting the corresponding engine operating regime.
Operational experience has shown that during a descent from 9000-
11,000 m with true speeds of 850-900 km/hr, at first a vertical speed of
8-10 m/sec must be maintained, then gradually decreased so that by
5000-6500 m, when the pressure in the cabin is constant (Figure 100] the
vertical speed is not over 5-6 m/sec. At altitudes of less than 5000 m, the
vertical speed can be increased to 10 m/sec. We will consider that the
thrust of the engines P acts in the direction of movement of the aircraft,
although as was stated above there is a certain angle 3 between force P and
the direction of movement of the aircraft. The lifting force Y is perpen-
dicular to the direction of movement of the aircraft, and the drag acts in
the direction opposite to aircraft movement.
For a stable descent, it is necessary that the aircraft weight component
G cos be balanced by force Y, and that force Q be balanced by the weight
component G sin and force P, i.e., that the following equality be fulfilled:
y=G cosQ; Q=P-fGsine.
°^ f %>fe"^^ Cho rd
Figure 99- Diagram of Forces Acting on Aircraft
During Descent: 1, Longitudinal axis of aii —
craft; 2, Descent trajectory; i? , Pitch angle;
0, , Flight-path angle; cfi, Rigging angle of
■ incidence; a, Angle of attack
The first equality is the condition for straight line movement, while the /145
second is the condition for constant velocity on the trajectory.
§2. Most Favorable Descent Regimes
In order to analyze the most favorable descent regimes from the stand-
point of fuel economy, let us use the formula Q = P + G sin 0, which char-
acterizes the condition of constant velocity. Let us analyze at first descent
with engines throttled.
We will consider that when the engines operate at the idle, the descent
occurs only under the influence of the component G sin 0, when Q = G sin 0.
Let us assume that the flying weight of the aircraft G = 33,000 kg, force
Q = 3000 kg with a quality of 11 and the flight speed is 810 km/hr. Then
sin = Q/G = 3000/33,000 = 0.091 and the angle of inclination of the
In order to maintain this angle 0, with a forward speed of
V = 810 km/hr (225 m/sec) it is necessary to maintain a vertical speed
Vy=V^ sin 6 = 225-0.091 =20.5 m/sec
As the flying altitude is decreased, the true speed of the aircraft will
decrease and, consequently, in order to retain the constant trajectory angle,
the vertical velocity component must be increased to 15-17 m/sec.
With this sort of vertical speed, the total descent time to the holding
altitude will be 10-12 min, and the total fuel expenditure 300-400 kg, the
descent range 120-170 km (considering the considerable decrease in vertical
speed involved at low altitudes) .
This method of descent is used when the cabin air pressure regulation can
provide normal conditions for crew and passengers. Another descent regime
is that in which the engine speed is maintained over the idle (in practice in
passenger aircraft the descent at idling regime is just being introduced) .
When this regime is used for the descent, the fuel expended is 400-500 kg
greater than in the regime described above, but satisfactory conditions are
maintained for passenger and crew. Table 10 shows the characteristics of the
descent regime with least expenditure of fuel for a TU-124 aircraft.
In comparison with the descent regime at the idle, the descent time is
almost doubled, and the range is increased by 50-100 km. The vertical
velocity components are selected from the condition of maintenance of a /146
constant pressure drop in the passenger cabin. The duration of the landing
maneuver (approximately from the region of the third turn, see Chapter IX) is
taken as 6 min (according to statistical data from scheduled flights) .
The next method is descent at the highest speed, in which piloting is
performed at the cruising (maximum permissible) M number or maximum indicated
speed. In this regime, the descent must be begun 270-300 km from the landing
point. The fuel expenditure during the descent is increased, since the
engines operate at a regime near the cruising regime for horizontal flight. /147
Table 11 shows the characteristics of the regime of descent at greatest speed
§3. Provision of Normal Conditions in Cabin During High Altitude Flying
The cabin of a passenger turbojet aircraft is sealed. In the cabin, the
temperature (20-22°C) , relative humidity and air pressure are maintained so as
to support normal vital activity of the crew and passengers during high
V , m/sec
An excess pressure over the atmospheric pressure is maintained in the
cabin (Figure 100). At altitudes between zero and 12,000 m, two pressure
regulation regimes are generally used:
a) The regime of constant absolute pressure, during which from ground
level to 4500-6500 m, a pressure of 760 nrai Hg is maintained;
b) A regime of constant pressure drop (difference between pressure in
cabin and atmosphere) , in which at altitudes over 4500-6500 m, the pressure in
the cabin is 0.5-0.65 kg/cm^ higher than the atmospheric pressure. With
Ap = 0.5 kg/cm^ at 8000 m, the cabin altitude is 1493 m, at 10,000 m -- 2417 m;
with Ap = 0.6, the cabin altitude at these altitudes will be 500-600 m lower.
Each of these regimes has a characteristic rate of change of pressure as
a function of altitude.
In the constant absolute pressure regime, the altitude in the cabin
remains unchanged during ascent and descent, equal to zero. Therefore, at
altitudes from zero to 4500-6500 m at any vertical speeds practically possible
(climb or descent) the rate of change of altitude in the cabin is equal to
zero. In the constant excess and variable absolute pressure regime, the rate
of change of pressure in the cabin is of essential significance for high
altitude passenger aircraft during a climb and particularly during a descent,
during which vertical speeds may reach 45-70 m/sec (in an emergency situation) .
At altitudes over 5000-6000 m, the vertical climbing speeds are usually much
less than descending speeds, 10-15 m/sec.
V , m/sec
t i me , m i n
The comfort o£ most passengers varies strongly with the rate of change in
barometric pressure. During rapid pressure changes (particularly during
descent) the passengers experience unpleasant and painful sensations in their
ears. Therefore, the rate of change of cabin pressure W , should be
W , = 0.18-0.20 mm Hg/sec, according to medical requirements. Maintenance
of W , within these limits at all altitudes over which pressure changes will
assure an even rate of pressure increase. The rate of change of cabin
pressure is equal to
W cab ^V^■^p„,
where V is the vertical rate of descent (climb) ;
Ap is the vertical pressure gradient of the atmosphere, mm Hg/m. For
H = 0, the gradient Ap^ = 0.09, for H = 8000 m -- 0.038 and for
H = 10,000 m -- 0.03 mm Hg/m.
. Hg /
Figure TOO. Pressure in
Sealed Cabin As a Func-
tion of Flying Altitude
(pressure drop Ap =
= 0.5±0.02 kg/cm^) :
1, Pressure in cabin;
2, Atmospheric pressure
This dependence can be used to deter-
mine the vertical rate of descent or climb
for any height, on the basis of the
condition of maintenance of normal
sensations of the passengers. For example,
let us determine the vertical rate of
descent of an aircraft for W
= 0. 18 mm Hg/sec:
For H =
„ __rcab 0,18
= 2 m/sec
For H = 10,000 m
= 6 m/^ec
Let us now determine the permissible "vertical speed" of the descent in a
passenger aircraft with sealed cabin at H = 10,000 m, if the cabin altitude is
2417 m and the vertical pressure gradient for this altitude Ap =
= 0.07 mm Hg/m: V = 0.18/0.07 = 2.5 m/sec. However, flying tests have shown
that an increase in the vertical velocity component at 10-12 km to 8-9 m/sec
and a corresponding increase in the vertical velocity of cabin altitude to
3-3.2 m/sec has almost no influence on the feelings of the passengers.
Therefore, the descent can be begun at 250-300 km from the airfield, in order
to provide normal landing maneuver.
An improvement in the valves of the cabin altitude system allows V to be
increased and therefore allows the descent to be initiated 100-120 km from the
landing point with the engines operating at the idle, which will provide a
savings of 350-600 kg fuel (the descent at the least fuel expenditure regime,
the idling regime, analyzed above) .
The permissible "vertical velocities" in the sealed passenger cabin of a
turbojet aircraft are presented in Table 12.
F 1 y i ng
V in cabin,
It follows from the above that descent from high altitudes should be
performed at a vertical rate of 8-9 m/sec down to 4500-6500 ra, then with any-
vertical rate required, as long as the permissible indicated speed is not
exceeded, since the pressure in the cabin will be made constant at 760 mm Hg.
§4. Emergency Descent
We have noted that in sealed cabins of turbojet aircraft the air pressure
is 640-540 mm Hg with a pressure drop Ap = 0.50-0.62 kg/cm^ (constant excess
pressure regulation regime) .
The change in the primary air parameters (pressure, weight density,
temperature and humidity) as a function of "altitude" in a sealed cabin is of
considerable significance for life support of man in flight. Of primary
significance is any change in partial oxygen pressure (p„ ) and its percent
The partial pressure of a gas included in the composition of any gas
mixture is that portion of the total pressure of the mixture produced by the
share of the gas in question. Oxygen enters the human organism, as we know,
through the lungs, the alveoli of which are covered by a network of blood
vessels. The penetration (diffusion) of oxygen through the walls of the blood
vessels into the blood can occur only if the partial pressure exceeds the
pressure of the oxygen in the blood. Similarly, removal of carbon dioxide
from the organism requires that the partial pressure of carbon dioxide in the
blood be higher than in the air in the alveoli of the lungs. Thus, whereas
the partial oxygen pressure at which normal gas exchange is assured under
surface conditions for the air inhaled is 159 mm Hg, this figure for alveolar
air is 105-110 mm Hg. The minimum permissible partial pressure of oxygen in
alveolar air, at which blood saturation of 80-85% will occur is 37-50 mm Hg.
This pressure corresponds to an altitude of 4.5 km, and this altitude cannot
be exceeded without special devices to increase the partial pressure
without oxygen starvation. This altitude is the physiological limit for
■ iiiiiiiiiiiiiiiiii iiiniiiiiiiiiiiiiiiiiiiii I 1 1 mil II nil mill I iiiii i ■inn
flight in nonpressurized cabins without oxygen devices. Oxygen starvation,
which causes so-called altitude sickness, may occur before this altitude,
since it depends to a great extent on the work performed by man. The
symptoms of altitude sickness are headache, sleepiness, decreased acuity of
vision and hearing, disruption of digestion and metabolism. These symptoms
begin to appear quite acutely beginning at 4.5 km due to the decrease in
oxygen supply to the cerebral cortex. It is difficult for the organism to
compensate for a decrease in the quantity of oxygen in the blood. Therefore,
the altitude zone from 4 to 6 km is called the zone of incomplete compensa-
tion. Above 6 km the critical zone begins, in which the disruption of mental
activity^ and functions of the organism becomes quite dangerous for survival.
In this zone, man loses consciousness and can only be saved by immediate
descent or supplementary oxygen supply. The critical zone ends at an altitude
of 8 km.
In case of a sudden sharp drop of pressure in the cabin (loss of cabin
pressure), oxygen starvation may occur. The time from the beginning of oxygen
starvation to loss of consciousness is called the reserve time. It must be
used to descend to an altitude providing sufficient oxygen concentration.
In case of a loss of cabin pressurization or in other cases (in
particular in case of fire on the aircraft) requiring a rapid descent, the
aircraft coimnander should decrease the flying altitude to 5000 m (safe
altitude) in 2.5-3 min or should perform an emergency landing.
An emergency descent should be performed at the maximum possible vertical
speed. This can be achieved by increasing the forward speed and the angle of
inclination of the trajectory. The greater the forward speed and the greater
the angle of inclination- of the trajectory, the greater will be the vertical
speed. However, the speed of an aircraft is usually limited at high altitudes
by the permissible M number, and at altitudes below 6000-7000 m by the
permissible indicated speed. Therefore, unlimited increases in forward
speed cannot be used, and the forward speed must be maintained within
The next possibility for increasing the vertical speed is to increase the
angle of the trajectory 0. The longitudinal forces must be equal during
descent at constant speed. It should be kept in mind that in a turbojet
aircraft during an emergency descent, the engines operate at the idle,
creating insignificant thrust. We can see from the equation P + G sin = Q
that sin = (Q - P)/G, i.e., the angle of inclination of the descent trajec-
tory (with constant aircraft weight) is greater, the greater the drag of the /151
aircraft. An increase in the drag of a turbojet aircraft can be achieved by
lowering the landing gear and spoilers. For example, during an emergency
descent, c of the aircraft is 0.024-0.026 for M = 0.84-0.86. Lowering the
landing gear increases c of the aircraft by 0.015-0.020. Lowering the
spoilers can increase c still more. In spite of the high flying altitudes
(9000-11,000 m) , the impact pressure reaches high values (for example, for
V = 900 km/hr at H = 10,000 in, q = 1300 kg/m^, while at 6000-7000 m with
V. J = 650-700 km/hr it is over 2000 kg/m^) , which makes it difficult to lower
and lock the landing -gear if they are raised with the flow, or to lower them
if they are raised against the flow. Therefore, in order to lower the landing
gear the indicated speed must be decreased by 40-60 km/hr. The loss of time
to achieve this is compensated for by the considerable increase in angle of
inclination of the descent trajectory and, therefore, the decrease in time
required for the emergency descent. A't the same time, raising the spoiler is
practically independent of the impact pressure.
Emergency descent of an aircraft can be divided into three main stages':
1) transition to descent with attainment of the maximum vertical velocity of
35-40 m/sec with landing gear up or 65-70 m/sec with landing gear down;
2) stable descent with these vertical velocities without exceeding the maximum
permissible M number at high altitudes or permissible indicated speed at low
altitudes; 3) bringing the aircraft out of the descent.
Energetic transition from initial cruising regime to the descent at
M = 0.78-0.80 is performed with an overload n = 0.6-0.55, and the control
should be performed using the overload indicator of the AUAP device
(chapter XI, §15). During this transition, V = 35-40 m/sec can be achieved
in 12-15 sec, with the M number increasing only to 0.82-0.84 (with landing
gear up). With a smooth transition with an overload of 0.9-0.8, the vertical
speed will only reach 25-28 m/sec after 35-40 sec, and the M number will be
approximately 0.85-0.86, i.e., the rate of increase in M number exceeds the
rate of increase in vertical velocity. If this mode of transition is used,
the aircraft may quickly reach the maximum permissible M number or exceed it.
If the transition is performed with n = 0.4-0.3 or less, it becomes difficult
to control the increase in vertical velocity, and the aircraft may reach
V > 35-40 m/sec and subsequently exceed the permissible M number. Therefore,
the transition to the descent should be performed with n = 0.6-0.55, which
(as will be seen below) corresponds to attainment of a vertical speed of
15-17 m/sec in the first 5-6 sec.
The second stage of the descent consists of maintaining a vertical speed
of 35-40 m/sec with landing gear up or 65-70 m/sec with landing gear down,
with the M number increasing to the maximum permissible value at the same /ico
time. The aircraft should continue descent at this M number down to 6500-
6000 m. The practically permissible M number is retained for 50-60 sec, then
decreases as the maximum indicated speed is reached. Subsequently , as
descent is continued at constant indicated speed, the M number drops (by
approximately 0.08-0.1 by 5000 m) , and the vertical speed decreases from
35-40 to 20-25 m/sec.
Flying tests have shown that it is not necessary to attempt to bring the
aircraft up to the permissible M number, but rather descent can be formed at
an M number 0.02-0.04 less than the permissible, since if the permissible
M number is exceeded, subsequent deceleration of the aircraft will sharply
decrease the vertical speed. It cannot be excluded that during the process of
a descent the velocity of the aircraft will exceed the permissible value
(either permissible M number or indicated speed). In these cases, it is
necessary first of all to halt further increase in M number, by slightly
decreasing the vertical speed (by 5-7 m/sec) , then once more decrease the
vertical speed by 5-7 m/sec, and when the M number reaches its permissible
value, to re-establish the constant vertical speed of 35-40 m/sec (or
65-70 m/sec with landing gear down) .
The third stage in the descent is a smooth transition back to horizontal
flight. This must be performed when the safe altitude is reached with an
overload n = 1.1-1.2, corresponding to a loss of 350-400 m altitude. The
transition from the descent (creation of n not over 1.2) is achieved by
observing the change in altitude, overload and vertical speed, not allowing
the maneuver to be performed in less than 300-400 m.
As we can see from Figure 101, the flying altitude of the aircraft with
landing gear up decreases by an average of 1000 m each 30-32 sec, and the
total time of descent is 2 min 30 sec-2 min 40 sec. With the landing gear
down, descent from 10,000 to 5000 m occurs in approximately 2 min. The
indicated speed gradually increases from the cruising speed (480-500 km/hr) to
the maximum permissible speed (700 km/hr) retaining this latter speed for
20-25 sec from 6500 down to 5000 m (landing gear up) .
The M number is increased from the cruising value of 0.78-0.82 to 0.85
(for this concrete case) which it retains for 50-52 sec, then decreases.
The vertical speed increases over 17-20 sec to a value of 35-40 m/sec
(landing gear up), then retains this rate down to 7000-7200 m, after which
(due to the attainment of an indicated speed of 700 km/hr, which must be
maintained by decelerating the aircraft with the elevator) it is decreased.
With the landing gear, the vertical speed reaches 65-70 m/sec and retains this
level for 50-60 sec.
The overload is decreased during 5-6 sec of the initial transition from /153
its initial value (n = 1) to 0.6-0.4, then increases to its initial value and
further (depending on the pilot's operation of the stick), remaining between
1.1 and 0.9.
The pitch angle r? varies from 2° (cruising flight) to -(7-8°) with
landing gear up or -(20-22°) with landing gear down.
The angle of inclination of the trajectory in a stable descent is
= i> + (j) _ a. For example, let us determine angle if the descent is
performed at M = 0.86 with V = 38 m/sec, where H = 8000 m, the weight of the
aircraft is 34 t, the wing setting angle <j) = 1°; we know from calculation that
for these conditions c = 0.171, a = 1°, q = 1885 kg/m^. Then
V = aM = 308-0.86 = 265 m/sec = 955 km/hr, and angle = !? = 8°, since
sm 8 = -^=—=0.143 and 0=8°;
&=e + a— <p = 8°-t-r — 1° = 8°.
In order to achieve a descent with landing geaf down with a vertical speed of /154
70 m/sec and a forward speed of 955 km/hr, angle = 15-16°.
Landing gear extended
Figure 101. Recording of Parameters During Emergency
Descent of Turbojet Aircraft: • , With landing
gear up from H = 10,000 m, M.^.^ = 0.78; , With
landing gear down and preliminary deceleration fr
H = 11 ,200 m, M. . = 0.8
in 1 1
The method of piloting an aircraft with landing gear up during an
emergency descent consists of the following. Before beginning the descent,
engines are set at the idle and, by moving the stick rapidly forward, the
pilot puts the aircraft in a descent. During this maneuver, the pilot must
check the indications of the variometer, overload indicator and M number
At the moment when V = 15-17 m/sec is attained, pressure on the stick
must be reduced, pulling it gently back so as to retard the increase in
vertical speed slightly. IVhen V = 25-30 m/sec is achieved, the stick must be
pulled back smoothly to retard the increase in vertical velocity still more,
gradually going over to a stable descent at a constant speed of 35-40 m/sec.
During the process of increasing V from 30 to 35-40 m/sec, the M number
indicator must be watched, to avoid exceeding the maximum permissible value.
Subsequently, a constant vertical speed of 35-40 m/sec is maintained using the
variometer, and the M number is not allowed to exceed the maximum permissible
until the maximum permissible indicated speed is reached (at approximately
6500 m) . When the maximum permissible indicated speed is achieved, the
descent is continued at this speed until a safe altitude is reached.
The load can be relieved using the elevator trimmer in the process of
stable descent when an indicated speed of 580-620 km/hr is achieved, so that a
pressure of 5-10 kg is maintained on the control stick. If the force is not
relieved by the trimmer, it will reach 50-60 kg. As the indicated speed
increases from 480-490 (beginning of descent) to 680-700 km/hr, the elevator
trimmer is moved away by 2.5-3°, and the deflection of the trimmer reaches
4-4.5° by the time an indicated speed of 700 km/hr is reached.
As the assigned altitude is reached, the aircraft is brought out of the
descent in such a way that it loses no more than 300-350 m altitude in the
maneuver. This corresponds to an overload of n = 1.16-1.2. At a vertical
speed of 5-6 m/sec, the engines can be transferred to the required regime.
Piloting the aircraft during an emergency descent with landing gear down
differs only slightly from the above. After the engines are shifted to the
idle, the landing gear control lever is moved to the "down" position, and the
aircraft is decelerated until the landing gear are completely down (at high
impact pressures, this may require 20-22 sec), after which the aircraft is
put into the descent by smoothly but forcefully moving the stick forward. Due
to the increase in drag resulting from lowering the landing gear, the overload
involved in the transition may be slightly less than in the preceding case
(the value may reach 0.3-0.4), since the acceleration of the aircraft to the /155
maximum permissible M number occurs somewhat more slowly.
When a vertical speed of 22-24 m/sec is reached, the pressure on the
stick must be decreased, and at V = 35-40 m/sec the rate of increase in
vertical speed must be decreased, and a vertical speed must be gradually
brought up to 65-70 m/sec.
Chapter IX. The Landing.
§1. Diagrams of Landing Approach Ziif.
The descent of an aircraft in the region of the airfield to the altitude
of circling flight is generally performed using the outer marker beacon
(0MB) or the entrance corridor beacon using the direction finder-range finder
system, the on-board and ground based radars.
During the process of the descent, the aircraft is guided to the airfield
so that the flying time in the region of the airport is 5-6 min. This allows
the fuel expenditure to be decreased (the aircraft flies for a short period of
time with landing gear down) , and decreases the flying time and cost of air
Therefore, the approach is either direct or uses the shortest path, in
which the aircraft is brought in in the region of the third turn (Figure 102) .
If the approach is direct, at 25-30 km from the airfield the aircraft descends /j-So
to 400-600 m and decreases its speed to the landing gear down speed. When
this altitude is reached, the landing gear are lowered at 12-15 km from the
0MB (this range is checked using the range finder or by commands from the
earth), and the flaps are lowered by 15-20°. The flaps are lowered completely
before entering the glide.
During a descending approach, the speed of the aircraft is decreased
in the region of the third turn during the process of descent to the circling
altitude, and the landing gear are lowered. The flaps are dropped by 15-20°
between the third and fourth turns. The fourth turn is performed with this
flying configuration, usually at 12-16 km from the runway, the flaps are
deflected fully and the aircraft follows the course to the runway at constant
altitude until it enters the glide path.
With forward movement speeds in the descent of 350-500 km/hr and landing
speeds of 200-250 km/hr, a jet aircraft will cover considerable distance
during the process of descent and speed reduction. Therefore, the extent of
the turns and particularly of the straight line sectors between turns will be
correspondingly increased. As a result, after the fourth turn the aircraft
will be at a considerable distance from the runway (12-16 km).
The inclination of the glide path is generally 2° 40 min-4°, as a
result of which the trajectory of the aircraft (after it enters the glide
path) is smooth. The glide path is entered at 7.5-8.5 km from the runway.
The 0MB is generally located 4 km from the runway, the boundary marker
beacon (BMB) at 1000 m from the runway. The altitude over the 0MB should be
200 m, over the BMB -- 60 m. For these flying altitudes, the vertical
velocity component of the aircraft should be 3-3.5 m/sec.
— 'I '
/ y//// //y //' //y
T 500 -a 500 m
Figure 102. Diagram of Approach to Landing (a) and
Gl ide (b)
§2. Flight After Entry into Glide Path. Selection of Gliding Speed
According to the norms of ICAO, the gliding speed during the descent on
the glide path should be 30% greater than the stall speed for the landing
configuration of the aircraft, i.e., V , = 1.3 V^ fwhere V_, is the stall
speed with flaps in the gliding position) .
As v/e can see from Figure 16, for a maximum flap angle of 38°, flow
separation on the wing begins at c =1.85. For a mean landing weight of 35 t
and a wing area of 110 m^, this corresponds to a stall speed
V = 14.4/35,000/110-1.85 = 190 km/hr.
Then the gliding speed is
V j=l,3I^s,= l,3-190^250 kn/hF.
Before the beginning of leveling off, gliding is performed at constant
speed, in this case 250 km/hr. With the standard angle of inclination of the
2° 40 min, the vertical rate of descent V = V , sin = 69. 5 '0.0466 =
= 3.24 m/sec (here sin 2° 40 min = 0.0466, V = 250 km/hr = 69.5 m/sec) .
Establishment of a constant gliding speed after complete lowering of
the flaps facilitates piloting, since it does not require a change in the
operating regime of the engines or a decrease in the speed from the moment
of entry into the glide path until the aircraft passes over the 0MB, BMB and
500-m mark, so that the pilot is less distracted from the instruments.
If the aircraft enters the glide path at 400 m altitude and 8 km range
from the runway (Figure 102) , flight to the 0MB in calm air (the aircraft
crosses the beacon at 200 m altitude) requires t = 200: 3.24 = 61 sec.
The difference in altitudes of flight over the 0MB and BMB is 140 m,
and the time of descent for this difference t = 140: 3.24 = 43 sec. The
flying speed of 250 km/hr corresponds to an angle of attack a = 5°
(Figure 16). Let us now determine, assuming (j) = 1°, the position of the
aircraft concerning the landing glide path, i.e., the pitch angle:
^ ^ -2° 40 min + 5° - 1° = 1° 20 min.
Thus, the aircraft axis has a positive angle with negative descent
angle 0. If, due to high mechanization of the wing (three slit flaps and
secondary control surfaces) the gliding speed is decreased (240-220 km/hr) ,
the pitch angle increases. Therefore, the flying time from the moment the
aircraft enters the glide path until it flies over the 0MB and BMB at lower
speeds is increased, and the pilot's reserve time increases. As a result,
the fourth turn can be formed closer to the end of the runway.
As the gliding speed is decreased at the same trajectory angle, the
vertical speed is decreased, and with the increasing angle of attack the
pitch angle increases, worsening the view from the pilot's cabin.
Let us analyze the engine operation regime required for gliding flight
of the aircraft .
With the landing gear down, flaps down and airbrake extended, the aero-
dynamic quality of the aircraft K = 5-6 and the gliding angle = 9-10°
(tan = 1/K = 1/5.5 = 0.183, ~ 10°), but in this case the engine thrust
should be near zero.
Actually, the aircraft descends along the glide path with engines
operating at angle = 2° 40 min. This angle corresponds to quality
tane tan2°40' 0.0466
For c =1.06 (angle of attack 5°, Figure 16), we produce c =0.19
(without airbrake) . From this value of c we must subtract the value of
coefficient c„ of required engine thrust, in order to maintain K = 21.5 where
-0.19=:Q.0493-0.19 = !0.14I(
Cy _ 1.06
This value of thrust coefficient corresponds to a thrust consumption
P = c qS = 0.141'300-110 = 4650 kg, i.e., 2325 kg thrust for each engine
(with a two-engine aircraft) . This thrust is several times greater than the
idling thrust (300-500 kg). If the airbrake is extended, the thrust must be
increased (to maintain the gliding angle unchanged, since c is increased to
c^ = -•-, - 0.226 = 0.0493 - 0,226 = |0.1 771;
p=0J77-300- 110 = 5840 kg
As we can see, the thrust is increased by almost 25%.
If after the airbrake is extended the engine operating regime is left
unchanged, the angle of inclination of the descent trajectory will be
increased to 4° 30 min and the aircraft may come down before the beginning of
the runway. In order to determine the new angle of descent, we must first
find the quality of the aircraft from the equation c = (1.06/K) - 0.226 =
/C^-.J ■5^ = 12.5,
and then find the descent angle
tane=-=----0 08 3nd e = 4W.
The effectiveness of the airbrake is quite high.
since as c is increased
the lift of the wing remains practically the same. Therefore, as the landing
gear are lowered the aircraft has no tendency to wing stall, but only shows a
change in the inclination of the trajectory.
§3- Stages in the Landing
The flight of the aircraft (descent) from 15 m (according to the ICAO
norms) consist of the following main stages: 1) gliding from 15 m altitude at
V = 1.3 V until leveling off is begun; 2) leveling off until the moment of
landing and 3) the landing run.
Figure 103 shows a diagram of the definition of required runway length
and a profile of aircraft flight from 15 m downward.
The total length of the horizontal projection of the trajectory of the
airborne sector and the landing run is called the landing distance. The
required runway length is determined for standard and design meteorological
conditions with the maximum landing weight of an aircraft and dry runway.
glide K- -v||ing,
- — required runway length ?=
landing dist x 1 .^3
Gliding -- straight
line flight of the
aircraft on a
at constant velocity.
Gliding is usually
performed at 250-
220 km/hr indicated,
with an angle of attack
a = 5-5.5° and
c = 0.95-1.1.
Profile of Descent of Aircraft
from H = 15 m
Pre landing gliding
is not gliding in its
pure form, since the
approximately 1800-2000 kg thrust each. This thrust is required to retain the
aircraft speed and retain good motor readiness in case it becomes necessary to
circle once more or for additional thrust to correct the landing pattern. If
the airbrake is extended, the engine operating regime must be increased by
5-6%, increasing the safety in case a second circle is required.
When gliding from 15 m to the height where the leveling is begun, the
aircraft travels 150-200 m. The vertical speed in the sector is 3-5 m/sec.
With the airbrake extended, the quality is decreased to 4.5-5, and the
angle of inclination of the trajectory can be increased when necessary to
9-11°. In this case, the length of the gliding sector from 15 m down
decreases to 100-150 m. The vertical speed can be increased to 8-9 m/sec.
Extending the fuselage airbrake creates pitching moment and facilitates
balancing the aircraft, since the flaps tend to create a pitching moment in
the opposite direction. The aircraft must be balanced so that slight pulling
loads are felt on the control stick at all times.
Leveling off. During leveling off, which begins at an altitude of
8-10 m, the movement of the aircraft is curved and the speed decreases. By
pulling the stick back, the pilot increases the lift, which becomes greater
than the weight component and therefore the trajectory is curved. In /160
practice, during leveling off the aircraft does not fly horizontally, but
rather at a slight angle to the ground (0.5-0.8°). In performing this oper-
ation, the pilot decreases the angle of inclination of the trajectory and the
vertical rate of descent to the point that a "soft" touchdown is provided.
This decrease in speed results from two factors: first of all, the angle of
attack is increased, increasing drag Q (for stable landing angles of attack
9-10°, the drag increases by 25-30%) and, secondly, before the beginning of
leveling off the pilot throttles back the engines and thereby decreases their
thrust. Leveling off is completed at an altitude of 1-0.5 m, so that the
touchdown occurs on the main wheels at landing speed with slight parachuting.
In order to retain lift during the process of leveling off, the angle of
attack must be increased to the landing angle of attack. During parachuting,
the lift is less than the weight of the aircraft by 25-30%.
When an aircraft lands with airbrake retracted, the length of the
leveling sector is increased, while if the airbrake is extended, due to the
better braking the length of the landing sector is decreased by 50-100 m.
During the leveling sector, the speed of the aircraft is decreased from
V ^ to V^ , . The length of the leveling operation depends on the difference
between these speeds. With a difference of 30 km/hr, it amounts to 350-400 m.
The greater the landing angle of attack (8-10°), the longer the braking of the
aircraft and the greater the length of the leveling sector. As a result, the
landing distance increases, in spite of the fact that the length of the run is
decreased slightly by landing at high angle of attack. As flying tests have
shown, it is more suitable to "brake" on the ground (during the run) than in
the air, when the aerodynamic quality is rather high (6-7) . This leads us to
the following conclusion: in order to avoid lengthening the holding sector
unnecessarily, landing should be performed with V^ , ~ ^ 1 "20 km/hr.
The run. The speed at which the aircraft touches the ground is called
the landing speed. It can be determined from the following formula:
where c ,, is the lifting coefficient at the moment the aircraft touches the
The run begins from the moment the aircraft wheels touch the landing
strip. The movement of the aircraft during this sector is straight and slow.
At first the run is accomplished on the main wheels, then by moving the stick
forward the pilot lowers the nose wheels. Most of the run occurs on three
points with a low angle of attack. On the polar curve, this corresponds to
the standing angle of attack 1-3° (Figure 65) .
Immediately after grounding, when the aircraft is rolling on two points, /161
the spoilers are deflected and wheel braking begins. Whereas at the moment of
landing coefficient c = 1.4-1.7, after the spoilers are extended, due to the
flow separation on the wing, it is decreased to 0.08-0.12. The lift
decreases sharply and complete loading of the landing gear wheels occurs.
It should be noted that at the moment the spoilers are extended a
negative pitch moment is acting on the aircraft and the pilot must push the
stick forward slightly to hold the aircraft at the landing angle of attack.
Extending the spoilers decreases the speed of the aircraft by 40-50 km/hr,
which causes the aircraft to tend to drop its nose rapidly, to which the pilot
must react by pulling the stick back to allow the nose wheel to drop smoothly.
Figure 104 shows an aircraft during the landing run with spoilers
extended and braking parachute out. During the process of the run, the
aircraft is decelerated by the drag of the aircraft and the friction of the
wheels on the ground. The slight engine thrust decreases this decelerating
The diagram of forces acting on the aircraft during the landing run is
the same as during the takeoff run (Figure 66) . The only difference is that
during the landing run the thrust P is considerably less than the sum of
decelerating forces F- and Q.
During the landing run, the summary braking force is defined as the
difference between decelerating forces and the thrust of the engines:
R, = Q + Fj- - P. As a result of the effects of the braking force, a negative
acceleration (i.e., deceleration) appears
_R^^ Q + Ff-P.
It follows from the formula that the greater the sum Q + F-, the greater
will be j . The friction force F_ depends on the coefficient of friction of
wheels with the surface of the earth f and the force of normal pressure of
the aircraft on the earth N. It has been determined by testing that for air-
craft with disk brakes and spoilers running on dry concrete f = 0.2-0.3
(considering braking) .
Force N depends on the landing weight of the aircraft and the lift:
N = G - Y. The force of friction can be expressed by the following formula:
At the beginning of the landing run, when the lift is only slightly less
than the weight, the force of friction will be low (low difference G - Y) .
For example, at 200-220 km/hr, the force of friction is 4000-5000 kg (for an
aircraft with a landing weight of 35-40 t) . At the end of the run, when the
lift is slight, the force of friction increases.
Figure 104. Aircraft During Run with Spoilers
Extended and Braking Parachute Out (a) and Diagram of
Opening of Spoiler (b) : 1, Inner spoilers; 2, Outer
spoilers; 3, Spoiler; k. Front flap; 5, Door; 6, Flap
The force of aircraft drag at the beginning of the landing run (when the
speed is near the landing speed, and angle of attack a = 9-10°) is rather
great (Q = 5000-6000 kg for the same weights) . This is facilitated by the
lowered flaps and the airbrake.
The landing distance (Figure 103) is the summary length of the sectors of
gliding, leveling and landing run. For aircraft with two engines in the tail
portion of the fuselage, the landing distance is 1000-1200 m, and the required
runway length (according to ICAO) is 1400-1700 m.
§4. Length of Post-landing Run and Methods of Shortening It
The kinetic energy of the aircraft at the moment of touchdown is
dissipated and absorbed by the work of the braking forces: the aerodynamic
drag, -the friction of the wheels on the surface of the runway, the drag of
braking parachutes, thrust reversal, etc. The dependences of these braking
forces on the speed of the run are shown on Figure 105. The unit of braking
force (drag force) used is the aerodynamic drag of the aircraft at touchdown.
For example, for the TU-124, at the moment of touchdown with flaps at 30° and
airbrake extended at 225 km/hr, c = 0.18, the aerodynamic drag Q = 4600 kg,
the parachute drag is approximately 5500 kg and the braking force of the
wheels is about 2500 kg. As the speed of the landing run decreases, the drag
force of the parachute and the aerodynamic drag of the aircraft drop sharply,
while the force of friction of the wheels increases. Thrust reversal of the
engines is practically independent of the rate of movement of the aircraft.
The length of the landing run of an
aircraft can be determined using the
■^^ m . 1 . r .
where j , is the mean acceleration of
braking (deceleration) of the aircraft
during the landing run, m/sec^.
As we can see from the formula, with
fixed landing speed the length of the run
can be decreased by increasing the mean
i — ~
iuu 1B0 v,KM/ hr
Figure 105. Nature of
Change in Braking Forces
During Post-landing Run
of Aircraft (calculated);
1 , Braking force;
2, Aerodynamic drag of
aircraft; 3, Drag of
h, Thrust reversal
During the first half of the landing run (Figure 105) the deceleration of
aircraft movement is achieved under the influence of all these decelerating
forces, after which the main role is played by the braking force of the wheels
and thrust reversal (if there is a thrust reverser on the aircraft) .
At the present time, braking wheels are equipped with special automatic
braking devices, the principle of operation of which is based on the usage of
the force of inertia of a flywheel rotating in parallel with the wheel.
If the wheel rotates without slipping, the flywheel in the automatic device
rotates in synclironism with the landing wheel. If the wheel begins to slide,
the flywheel introduces an acceleration and, working through a special device,
interrupts the supply of pressure to the brake, as a result of which the
braking force on the wheel is decreased. After the rotating speed of the
wheel is increased once more and synchronism is established between rotation
of wheel and flywheel, the pressure to the brakes is increased to the required
level and the wheel is once more braked. In operation, this cycle is usually
repeated quite rapidly and actually the pressure in the brakes never decreases
completely. Thus, this device provides optimal braking, pumping at the
boundary of sliding-^ . When this device is turned on, the pilot immediately
provides full pressure in the brakes (depresses brake pedals completely) .
Smoothly depressing the brakes, as is recommended for nonautomatic
braking, in this case only increases the length of the landing run, since the
maximum braking regime will not be used.
The usage of automatic brakes has allowed the length of the landing run /164
to be decreased by an additional 20-25%. The service life of the pneumatic
system has also been increased. The mean acceleration of automatic braking is
1.7-1.8 m/sec^ (disk brakes). In aircraft with spoilers opened at the moment
of touchdown, the effectiveness of the brakes is even greater and
j = 2.25-2.5 m/sec^. For example, in an aircraft with spoilers
(j =2.25 m/sec^) with a landing speed of 216 km/hr (60 m/sec) , L = 800 m.
For the TU-104 aircraft (no spoilers) with V^ , = 240 km/hr (66.7 m/sec) with
an average braking acceleration of 1.3 m/sec^ (drum brake) the landing run
length is 1700 m. For the TU-104 with disk brakes (with an average acceler-
ation of 1.55 m/sec^) the landing run length is 1430 m.
Even greater braking acceleration (drag) can be produced by releasing a
braking parachute. For example, if the parachute is open at 225-215 km/hr,
the drag is increased by 4600-4900 kg (TU-124 aircraft) .
Figure 106a shows a diagram of the usage of a braking parachute. After
touchdown, a button is pressed dropping the parachute from its container
through hatch 1. After this, the pilot chute pulls the braking chute out,
creating resistance to the movement of the aircraft. The parachute is
connected to the aircraft by cable 3 through catch 2. At the end of the run,
the braking parachutes are disconnected. Braking parachutes 4 are
strip type, and the strength of the lines and canopy is sufficient for run /165
speeds of 260-230 km/hr. In a strip type parachute, the air partially passes
through the canopy and therefore for this type of chute Ac = 0.25-0.55 (for
an ordinary parachute Ac = 1.2-1.3). For example, one foreign braking
parachute with a canopy diameter of 9.76 m and Ac =0.55 creates a braking
'■ A. v. Chestnov, Letnaya Ekspluatatsiya Samoteta [Flying Operation of Air-
craft], Voyenizdat. Press, 1962.
force of 17.25 t at 296 km/hr (military transport aircraft).
The length of the landing run on an ice covered runway can be reduced by
30-40% by using a braking parachute. Under these conditions, its effective-
ness is particularly noticeable. However, the less the speed, the less the
effectiveness of the parachute. For example, the braking parachutes on a
TU-104 decrease the run length by 25-30% (wet or ice covered strip). Thus,
under standard conditions for a landing weight of 58 t, the runlength is
1730 m, while the usage of the parachute reduces this figure to 1250-1350 m.
The braking force is 10-14 t.
Opening for gas
■^-^.v . escape _
Figure 106. Usage of the Braking Parachute (a) and
Diagram of Installation and Operation of Thrust
Reversers (b) on Two External Aircraft Engines:
1, View from rear, reversed flow incl i ned by 20° from
vertical; 2, Apertures for gas outlet directed at
angle opposite to flight; 3, At moment of touchdown,
reverse doors closed, during braking they direct gas
in direction opposite movement. During taxiing,
doors set in intermediate position.
One defect of this method of reducing the run length is the fact that
with a side wind stronger than 6-8 m/sec at an angle of over 45° to the runway,
the parachute will be deflected from the axis of the aircraft and will tend to
turn the aircraft into the wind. As the side wind increases in speed, the
probability of rotation also increases. However, even in this case it is
recommended that the braking chute be used during the first half of the
landing run, being extended immediately after touchdown (in practice with a
delay of ^-7 sec) . Another defect is the fact that the discarded parachute
must be rapidly removed from the runway, transported, checked and packed. The
service life of a braking parachute (with an average acceleration of
1.55 m/sec^) is 40-50 landings. Calculation of the drag produced by the
parachute is performed using the formula
■ ■^^Hiiiiii ■ ■iinn^^^^Hiii^^iiiHiillHllllllllllll I I I I I mil III I Hill I II I
Q =Ac^ Sq,
where Ac is the drag of the parachute related to the wing area of the
S is the wing area;
q is the impact pressure.
For example, for the braking parachute of a TU-124 with S = 40 m^,
c = 0.54 (S = 105.35 m^) :
X par ^ ■'
Ac^„^= c arw^ ==0.205.
par s 105.35
Ejection of the braking parachute at lower speed is less effective. At
the end of the landing run, due to the decrease in speed and the angle of
attack, which will be equal to the parked angle, force Q is practically equal
to zero. It is considered that in the process of the entire landing run, an
average braking force acts on the aircraft, creating a average negative
The greatest value of negative acceleration is achieved after the braking /166
parachute is extended and amounts to 4.4-4.2 m/sec^.
Increasing the landing speed by 5% (from 210 to 220 km/hr) increases the
length of the landing run by approximately 10%. Therefore, a decrease in
landing speed is the most effective means of decreasing the run length. An
increase in j , by the usage of spoilers and a braking parachute or thrust
reversal of the engines can significantly shorten the landing run.
When the engine thrust is reversed, the reaction jet is directed forward
and exits upward and downward at an angle to the horizontal. For example, in
the two outboard engines of the English "Comet" turbojet aircraft, the reac-
tion jet exits upward and downward at 45° to the horizontal.
The reverser (the device which deflects the flow) is rotated at 20° to the
vertical, in order to direct the jet away from the fuselage and landing gear
(Figure 106 b) .
With sufficiently rapid movement of the aircraft, the jet will be
deflected rearward and will not enter the air intakes, while at very low
speeds or at rest of the aircraft the stream will move far forward.
The operating time of the reverser in a landing is generally not over
15 sec. The doors of the reversing device are operated pneumatically. The
reverser is put in operation by moving a special lever forward. The throttles
controlling the outboard engines must first be put in the idle position and
lifted. The effectiveness of thrust reversal is decreased with decreasing
However, when necessary thrust reversal can be used until the aircraft
comes to a complete stop.
Thrust reversal should be applied the moment the aircraft touches the
runway. The maximum reverse thrust theoretically is 70% of the forward
thrust, but in practice only about 50% is realized.
The usage of thrust reversal makes it possible to decrease the landing
run length by 20-25%. Also, in the "Comet-4B" aircraft the size of the flaps
is increased and their angle of deflection is increased to 80°, greatly
reducing the landing speed.
In aircraft with engines located in the wing and near the fuselage, the
usage of thrust reversal is difficult due to the thermal effects of the
reversed jets on the fuselage. It is easiest to use thrust reversers on
engines mounted on pilons, as on the Boeing 707, DC-8, etc. If there are
four engines mounted on the tail of the fuselage, the reversers are installed
only in the outboard engines.
As was noted, in addition to braking parachutes, motor switch off during
the landing run, and thrust reversal, spoilers and airbrakes are also used.
The spoilers are plates which can be extended or deflected, mounted on the
upper surface of the wings. One, two or three spoilers can be used on each /167
The spoilers are extended after the aircraft wheels touch the runway. By
separating the flow from the upper wing surface, the spoilers decrease the
lifting force sharply and create considerable additional drag.
The graph on Figure 107 shows that with the spoilers closed the aero-
dynamic quality of the aircraft decreases from 6 to 4.4 upon transition from
the landing position (a = 10°) to the landing run position (a = 1°); opening
of the spoilers during the run decreases the aerodynamic quality by an
additional factor of 4 (from 6 to 1.5).
Extending the spoilers has approximately the same influence on the
dependence c = f (a) .
§5. Length of Landing Run As a
Function of Various Operational
The length of the landing run is
essentially influenced by the aircraft
weight, condition of the runway,
direction and speed of wind, air
temperature, etc. The length of the
landing run also depends on the
actions of the pilot in control of the
The weight of the aircraft
influences the length of the landing
run primarily through the landing
speed. As the weight of the aircraft
is increased, the square of the
landing speed is also increased and
consequently the length of the landing
run is increased to the same extent.
For example, with landing weight of
30,000 kg, the length of the landing
run under standard conditions is
930 m, whereas with a landing weight
of 32,000 kg, i.e., increased by
1.065 times, the run length is
increased by the same number of times and will be 930 '1.065 = 990 m.
Thus, if the aircraft weight is increased by 6.5%, the run length will be
increased by the same factor.
The temperature of the surrounding air influences the run length
primarily through the density. As the temperature is increased with unchanged
pressure, the density of the air is decreased. 2 If the temperature is
Figure 107. Coefficient c As
a Function of Angle of Attack
and Polar Curve of Aircraft
During Landing (flaps down,
Airbrake and Spoilers
increased by a certain factor, the value of
is increased by the same
Thus, if the temperature is increased by 5% over the standard
temperature, V^, will be increased by approximately the same percent.
A decrease in density leads to a decrease in the drag Q during the run.
Also, during the run the engines create a slight thrust and as the temperature
is increased, this thrust is decreased, which helps to reduce the run length.
If we ignore the influence of temperature on drag and thrust, we can approx-
imately consider that an increase in temperature of 5% (for example from 15 to
30°C (from 288 to 303°K) will result in an increase in run length of
It should be noted that under conditions other than the
standard conditions, the landing speed indicated by the instrument (the broad
arrow) will be the same as at standard conditions, since with a change in air
density the velocity indicator decreases the indicated speed due to methodic
error. The fine needle of the indicator shows the true speed in this case.
The influence of head winds and tail winds on the length of the landing
run is the same as this influence on the length of the takeoff run.
The braking effect is always greatest with the maximal speeds of
utilization of spoilers and parachute. Therefore, a delay in using the
spoilers of 1.5-2 sec increases the run length by 100-150 m, while ejection of
the parachute at 180-140 km/hr decreases its braking effect by 35-50%. The
wheel brakes should be applied immediately after the spoilers are extended,
i.e., at 250-220 km/hr.
§6. Specific Features of Landing Runs on Dry, Ice or Snow Covered Runways
At the present time we still do not have sufficient data on methods of
determining the effect of braking on wet or snow covered runways.
In spite of the variety of means of braking, the principal means remains
the disk wheel brakes. It has been established that when landing on a dry
concrete runway, about 70% of the energy of movement of the aircraft is
absorbed by the brakes, and 30% by aerodynamic drag of the aircraft (usage of
flaps and airbrakes) . When landing on a wet runway, only about 50% of the
kinetic energy is absorbed by the brakes, or if the tires are worn -- even
less. The wheel brakes have an important role to play during a landing run if
flight is terminated at speeds less than the separation speed by 15-20%, in
which the spoilers and landing parachute are less effective. The pressure in
the tires has a great influence on the effectiveness of braking: the less the
pressure, the greater the contact area and the more reliably the brakes
At the present time, the runway length required for aircraft operation is
determined either on the basis of the condition of the provision of safety of
interrupted or extended takeoff (see Figure 71) , or from the conditions of the /169
conditions of the landing characteristics of the aircraft (see Figure 103) .
These characteristics are generally calculated for a dry runway surface.
Hov^ever, at most airports due to climatic conditions over one third of the
year or perhaps even more the runway surfaces are moist, snow covered or
frozen. Statistics show that on the world scale, one landing of twelve is
performed on a wet runway^ .
^ Zaruhezhnyy Aviatransport , (Foreign Air Transport) No. 7, ONTI GOSNII GA
I [Technical Information Department, State Scientific Research Institute for
Civil Aviation], 1965.
The experience o£ operation of domestic turbojet and turboprop aircraft,
as well as data from foreign practice indicate that the presence of slush (wet
snow, water) on runway surfaces has the following negative influence on the
design of aircraft and landing operations : 1) additional drag appears as the
slush strikes the aircraft, particularly in the case of aircraft with heavy
landing gear; 2) the danger arises that liquid may enter the engine air
intake; 3) controllability of the aircraft is reduced; and 4) the landing
run length is significantly increased.
Pavements for runways include concrete, asphalt, etc. On a moist or viret
runway, the wheel roll drag increases, but the coupling force between wheel
and runway during braking decreases (in comparison to dry pavement) . This
results in an increase in the landing run length of the aircraft. This
increase is so great that in many cases the length of the runway may be
insufficient to complete the landing ixin.
A moist runway is understood to be the condition in which the pavement
is moistened with water (after rain) , while a wet runway means that there is
a layer of water on the runway 2-3 mm thick. Tests performed in the USA
showed that with a certain thickness of water on the runway and with certain
parameters of the tires, the critical speed can be reached at which the
tires are completely separated from the surface of the road by hydrodynamic
forces created by the liquid between the tire and the surface of the runway
(Figure 108 a) . This speed is called the skidding speed or speed of hydro-
The effect of aquaplaning significantly increases the landing run length
on a wet runway. Investigations have shown that aquaplaning arises at speeds
averaging over 160 km/hr. When this occurs, the contact between wheels and
pavement is lost and a flim of water appears between them. This results in a
loss of effectiveness of brakes and makes it difficult to maintain the
direction of the landing run. The phenomenon of aquaplaning is explained by
the fact that a hydrodynamic force acting on the surface of the pavement
arises as the aircraft moves over the runway. When its vertical component /ivn
becomes equal to or greater than the weight of the aircraft, contact of the —
wheels with the runway is lost.
The graph on Figure 108 b was produced theoretically and confirmed
experimentally. Using this graph (with known pressure in the tires) , we can
establish the limiting speed, above which usage of the wheel brakes during a
landing on wet surface is useless, or even dangerous in case of a strong side
wind, so that only aerodynamic brakes should be used. As soon as the speed
drops below the aquaplaning speed, the wheel brakes can be used.
At the moment the brakes are applied, a friction coupling force appears
between aircraft wheels and runway. In some cases braking may result in
wheel lockup (100% skid) i.e., a situation in which the movement of the
aircraft with nonrotating wheels (skid) causes the force of friction to
decrease, increasing the length of the landing run. The interaction of the
braking wheel with the runway surface is generally evaluated by the coupling
coefficient or coefficient of friction, equal to the ratio of the tangential
braking force to the normal loading on the wheel .
a) Di recti on of
V^\ '' S. '\V\\\
"^ s \s \\\s
Critical speed for
aircraft in quest
/ Z 3 U 5 B 1 2
Pressure in tires, kg/cm
Figure 108. Formation of Hydrodynamic Lifting Force As
Wheels Roll Along Wet Runway (a) and Aquaplaning Speed
As a Function of Pressure and Tires (b) : 1-2, Hydro-
dynamic lift and drag
On a clean, dry surface, the coupling coefficient of the tires is quite
high and, if the rubber does not melt or burn due to the high temperature at
the point of contact with the runway surface, this coefficient may vary
between 0.7 and 0.8 depending on the tread profile (dry concrete). As the
speed of the aircraft is increased, the coefficient decreases by 2-3 times.
Therefore, the mean value of coupling coefficient for a dry concrete
runway is 0.15-0.25; for a moist runway this figure is 0.1-0.21 and for a wet /■*■"■*-
runway, about 0.21-'-. For an asphalt runway (according to the data of the
State Planning Institute and the Scientific Research Institute for Civil
Aviation) 2, the coupling coefficient for all of the pavement conditions
analyzed above is somewhat higher: from 0.33 to 0.23; for snow covered cement
and asphalt pavements it is 0.3-0.25. Therefore the calculated landing run
length of an aircraft on these pavements is 15-20% less.
When landing on an ice covered runway, the effectiveness of the brakes is
sharply decreased, by an average of 25-30% in comparison with a landing on a
dry, concrete runway. Due to this, it is generally recommended that a braking
parachute be used, that one or two engines be shut down, etc. It is known
that rapid dropping of the front wheel onto the runway after touchdown creates
the best conditions for braking. However, as a rule, this method is most
suitable for a dry runway pavement, since on wet pavement, frozen or
^ Chestnov, A. V., Letnaya Eksptuatatsiya Samoleta [Flying Operation of the
Aircraft], Voyenizdat. Press, 1962.
2 GPI and Nl IGA.
snow covered pavement, the braking effect of the wheels is reduced. Under
these conditions, we must keep in mind the fact that running with the front
wheel up creates additional aerodynamic drag, which is the main braking
effect during this portion of the run. It is particularly difficult to
perform a landing (or takeoff) on a runway covered with wet snow. Experience
has shown that a layer of wet snow 25 mm thick increases the takeoff run
length by 60%, and that a layer 75mm thick makes a takeoff impossible.
The maximum permissible depth of a layer of liquid or water has been
experimentally established as 12.7 mm. This depth will require an increase in
takeoff run length of 20-30%.
§7. Landing with Side Wind
The side wind means the wind velocity component directed perpendicular to
At the present time, landings with side winds are made by the method of
course lead, i.e., drift of the aircraft is compensated for by creating a
certain lead angle e in the course of the aircraft after exit from the fourth
turn (Figure 109). If the course of the aircraft is changed by angle e,
determined from the relationship tan e = W/V , the ground speed V will be
directed along the runway. Thus, if V = 250 km/hr, while W = 10 m/sec, the
lead angle e = 8° . However, during leveling off and holding the speed of the
aircraft will decrease and the initial lead angle will become too low; the
aircraft will begin to drift off of the runway. Therefore, at the moment of
touchdown, the lead angle must be increased by approximately 1-1.5°.
The crew should have good visibility from the cockpit at lead angles of /172
10-15°, which are required with a side wind above 15 m/sec.
When drift is compensated for by a variation in landing course, the
longitudinal axis of the aircraft does not correspond to the direction of
movement, and flight is performed without slipping or bank. At the moment of
touchdown, the control wheel should be turned in the direction of the drift,
rotating the aircraft along the runway by lead angle e. If when this maneuver
is performed the longitudinal axis still makes a certain angle with the
direction of the runway, side force Z will act against the wheels, tending to
rotate the aircraft along the runway, since it is applied behind the center of
gravity of the aircraft; however, this effect is not dangerous for the landing
organs. As we can see from Figure 110, the nose wheel presents no moment,
since it is oriented freely along the direction of movement while the side
friction force on the main wheels creates stabilizing moment, tending to
rotate the aircraft to line up with the runway. With a side wind, gliding
should be performed at higher speeds (10 km/hr higher) , and the landing speed
should be 5-10 km/hr higher than the normal recommended speed. The pilot must
control his aircraft on the approach to the landing strip carefully, being
sure not to level off high or touchdown hard. The front leg must be lowered
immediately after landing in order to avoid zooming and to maintain the
direction from the landing run using the control wheel. The control stick
should be pushed forward to the stop in order to bring the nose wheel down to
When landing with a side wind, the length of the landing run is increased
by 10-15%. The maximum permissible value of side wind component (90° to
runway axis) is 12-15 m/sec. In case of a large rotational moment, the down-
wind engine may be switched off, the braking parachute can be released, thrust
reversal and braking can be used.
§8. The "Minimum" Weather for Landings and Takeoffs
The takeoff- landing characteristics of an aircraft determine the
limiting meteorological conditions ("minimum weather") for which operation of
the aircraft (takeoff and landing) can be permitted.
The conditions include: a) minimum ceiling; b) minimum visibility at
runway level; c) minimum lateral component of wind speed W .
The minimum ceiling determines the flying altitude to which the aircraft
should come down out of the clouds and clear visibility of reference points on
the ground or runway lights should be established. At this altitude, the crew
can guide the aircraft down on the landing line visually. For turbojet
aircraft landing at airfields equipped with ILS, with a glide path angle of
2° 40 min, the minimum cloud cover ceiling is 60-100 m.
The minimum visibility is considered the range at which the crew of an
aircraft begins to see reference points on the ground and the beginning of the
runway during the daytime, or landing lights and the illuminated runway
surface at night. This range should be sufficient to make it possible to
correct inaccuracies in aircraft course and separation from runway axis. The
accuracy of guidance of the aircraft relative to the center line of the
runway depends on the accuracy of output of course data by on-board and ground
base apparatus and the precision of piloting according to the indicator on
board the aircraft. Experiments performed by GOSNII GA-^ have established that
for passenger jet aircraft the mean value of total deviation from the runway
axis is *60 m. Coming down out of the clouds with this amount of error, the
pilot must correct the error with two sequential turns (Figure 111) . During
this time, the aircraft continues to descend on the glide path, generally
between 2° 40 min and 4° (the higher value for airfields with difficult
approaches). The time required to correct lateral deflection is influenced
considerably by the inertia of the aircraft, its delay (4-5 sec) to movements
■*■ S. M. Veger , Proyektivovani-ye Passazh-ivski-kh Reakt-ivnykh Samotetov [Design
of Jet Passenger Aircraft], Mash inostroyen iye Press, 196^.
of the control organs and the characteristics of lateral and transverse
stability. Furthermore, an additional 2-3 sec is required for crew reaction
from the time when the runway can first be seen. Therefore, it is required
that upon approach to the BMB or after flying over the BMB the crew of the
aircraft must be able to see the beginning of the runway from the point of
beginning of leveling off down to the touchdown (which in practice is
250-300 m from the beginning of the runway) . Minimum visibility is then
800-1200 m, or 1500 m for night landings.
Thus, the transition to visual flight (exit from the cloud cover at
60-100 m for a glide path angle of 2° 40 min) occurs at 1250-1500 m from the
beginning of the runway and during the subsequent 6-7 sec of flight (240-
250 km/hr velocity) the crew must have a clear view of the runway, the point
of beginning of leveling off and the point of touchdown. During this time,
the pilot can perform course maneuvers if the aircraft is coming in at an
angle, completing his maneuvers by the time he reaches an altitude of 40-50 m
(at 600-800 m from the runway) . Below an altitude of 50 m, it is forbidden
for a jet aircraft to pull up for a second circle. This altitude corresponds
approximately to flight over the BMB, and the crew should take all steps to
assure a normal landing from this point.
Figure 109. Elimination
of Landing Drift by
Course Lead Method
(f 1 ight wi th leading
Figure 110. Diagram of
Landing Run After Touch-
down with Lead Angle e
^'■.- ■,'.-.■ '^'
^ 2°^d'-^ — ~ -^ ^ runwa y^
2-j " maneuver
Figure 111. Determination of "Minimum Weather'
With lateral deviations of ±60 m and a gliding speed of 250-240 km/hr,
the required ground length to bring the aircraft over to the landing line is
800-900 m. If the aircraft comes out of the clouds at 100 m altitude and
1800-1900 m range from the runway and the pilot, upon seeing the runway,
decides to turn the aircraft, he can complete his maneuver at 600-700 m from
the runway and bring the aircraft onto the landing course. With greater
deviations (70-100 m^) the required ground length is 1000-1200 m and the pilot
will not be able to bring the aircraft onto the course line and perform his
landing in the space available. Therefore, the radar controller guiding the
aircraft into a landing, upon determining this abnormal deviation of the
aircraft from its course, should forbid the landing (before the aircraft gets
down to 50 m altitude) and require the aircraft to go into a second circle.
The "minimum weather" is established not only from considerations of
safety of landing of the aircraft under poor weather conditions, but also from
considerations of takeoff safety. As was stated above, the height at which
the aircraft flies over the BMB in case of extended takeoff with one non-
operating motor is 20-25 ra. If the height of obstacles in this flight sector
is not over 11-14 m, there is no limit on the ceiling. Horizontal visibility
should be at least bOO-diQQ m. This quantity is determined as follows.
During a climb after takeoff, the pitch angle i? = 6-8° (depending on the
angle of the climbing trajectory 0). The angle of view downward from the
crew's cabin for modem aircraft is 15-20°.
After takeoff at 60-70 m altitude (when the landing gear and flaps are
raised) the crew should see the runway or orientation points on the surface
such as approach lights (in order to maintain the takeoff course) at least
400-500 m in front of the aircraft. The additional visibility reserve due to
the slower reaction of the pilot is generally 2-3 sec, corresponding to an
additional 200-300 m. Thus, the minimum visibility during a takeoff should be
§9. Moving into a Second Circle
An aircraft may move into a second circle during any stage of the landing
approach, including the leveling off. High power reserve makes it possible to
move off into a second circle even with one motor out of operation (TU-104,
TU-124, TU-134) .
The decreased pickup of turbojet engines does influence the behavior of
the aircraft at the moment the transition is made to the second circle. The
problem is that the time required for the engine to shift from the idling
regime (300-600 kg thrust) to the nominal thrust regime or higher is IS-
IS sec, while in practice after 6-7 sec, i.e., after the throttle is placed in
the "maximum thrust" position, the engine thrust reaches a value sufficient to
provide not only horizontal flight, but some climb. On the basis of this, a
unified method of piloting in case it becomes necessary to make a second
circle has been developed (by Candidate of Technical Sciences M. V. Rozenblat) .
After deciding to enter a second circle, the pilot sets the throttle to
the maximum position. If the airbrake has been extended, its switch is
shifted to the "retract" position. The aircraft is brought out of the
descent and the speed is retained unchanged until the aircraft begins to /176
climb. Six to eight sec after the throttles are pushed into the maximum
position, the engines will develop thrust equal to 75-80% of the maximum
(Figure 112, point 2), which will overcome the drag of the aircraft with some
excess power available. When the available power exceeds the required power,
the aircraft will begin to climb.
Vlhen necessary (for example with increased vertical descent rate) in
order to decrease the rate of descent, immediately after the engines are
shifted to the maximum regime the flight speed can be smoothly reduced by
10-15 km/hr, but never below the established gliding speed.
After the aircraft is shifted into a climb and the engines reach the
maximum regime, the landing gear are brought up, causing the flying speed to
increase sharply. When a safe speed is achieved and an altitude of 80-100 m
is reached, the flaps are raised, and the engines are shifted to the nominal
or cruising regime. The landing gear should not be raised until the engines
reach a regime providing sufficient thrust for flight, since the drag of the
aircraft is increased when the landing gear storage bay doors are opened
causing the rate of descent to increase. The graph of Figure 112 shows that
the aircraft continues to descend until the engines reach the required regime;
when the vertical velocity component V = 3.5-4 m/sec, the additional descent
will be 15-20 m. With V = 5-7 m/sec, the additional descent will be 30-40 m
if the speed is retained the same, or 20-25 m if the flight speed is decreased
by 10-15 km/hr. Therefore, the lowest safe altitude for the decision to make
a second circle with landing gear down, flaps in the landing position and
airbrake on is usually 50 m. With the additional descent of up to 30 in, an
altitude reserve is thus guaranteed.
If the speed of
the aircraft is
decreased by 10-
15 km/hr in the range
of gliding speeds
240-260 km/hr, the
resulting from kinetic
energy is 18-25 m.
70 Z, sec
Figure 112. Change in Altitude and Flight
Speed of TU-124 Aircraft upon Transition to
Second Circle from Altitude of 75 m (average
weight 33 t, 6^ = 30° and 6,^^ = 40°) :
1, Moment of throttle shift and beginning of
retraction of airbrake; 2, Moment of achieve-
ment of 75-80^ maximum thrust by engines;
3, Moment of transition of engines to takeoff
regime and beginning of raising of landing
gear; 4, Beginning of raising of flaps
Chapter X. Cornering
§1. Diagram of Forces Operating During Cornering
0£ all of the curved trajectory maneuvers in the horizontal and vertical
planes, the transport aircraft is permitted to perform only the cornering
maneuver -- flight in a curved trajectory in the horizontal plane with a
360-degree turn. A portion of a cornering maneuver is called a turn. A
stable cornering maneuver without slipping is considered proper.
In order to perform
cornering it is necessary that
an unbalanced force act on the
aircraft, curving the traject-
ory, and directed perpendic-
ular to the trajectory
(Figure 113). This force is a
component of the lifting force
Y sin Y (where y is the bank
angle) , produced when the
aircraft is banked. This
force is called centripetal;
it results in the appearance
of a force equal and opposite
to the centrifugal force:
Figure 113. Forces Acting on Aircraft
During Cornering: a, Proper cornering;
b, Cornering with outward slip (nose
of aircraft deflected toward interior
_ = m — ,
As the banking angle is increased in a proper turn, the lifting force
must be increased so that its vertical component Y cos y continues to balance
the weight of the aircraft.
The forces acting on the aircraft during a horizontal turn should satisfy
the following equalities
If Y is expressed through the overload n = Y/G, then
This formula shows the relationship between overloading, which must be used to
perform the horizontal turn and the banking angle y (Figure 114) . As we can
see from the graph, in order to perform a horizontal turn at y = 60°, we must
create n =2.
In passenger aircraft, the bank angle is
usually set at 20-30°, which affords the
During an approach to landing under instru-
ment flight rules, the bank cannot exceed 15°.
With most modern aircraft, horizontal turns
are performed using the ailerons alone, almost
without using the rudder, with the aircraft
"itself" selecting an angular turning rate so
that there will be no slippage. This has become
possible due to the high degree of directional
stability, which greatly facilitates maintenance
of so-called "coordination," i.e., a combination of operations of the ailerons
and rudder for which the velocity vector remains in the plane of symmetry of
the aircraft and no slipping occurs^.
Figure ]]h. Over-
load As a Function
of Banking Angle
§2. Cornering Parameters
Cornering parameters include the radius of the horizontal turn, time of
the turn, angular velocity of the turn, etc.
The following formulas are known for the radius and time of a horizontal
^ M. L. Gal lay, "Lateral and Directional Stability of the Aircraft," Letchiku
o Prakt-ioheskoy Aerodinamike [Practical Aerodynamics for the Pilot],
Voyenizdat. Press, 1961.
&9L= — ^oJ^ ; /cor 0-64-^^.
r= — ^^— — ' 'CO
where V is the speed during the cornering maneuver;
g is the acceleration of gravity; /179
n is the overload;
Y is the bank angle of the aircraft.
We can see from the formula that the radius of the turn depends strongly
on the flight speed, increasing rapidly with increasing speed. The radius of
the horizontal turn can be decreased by increasing the overloading, i.e., by
increasing the bank angle of the aircraft.
During cornering, the aircraft has an angular velocity of
Let us calculate the radius of turns performed during the landing
approach around a large, rectangular course (y = 15°, tan 15° = 0.268).
If the bank angles and the turns are greater than 15°, the maneuver-
ability of the aircraft increases and the landing approach time decreases (the
reserve of pilot's time increases).
For all aircraft, the first turn in the approach to landing begins
according to the diagram at 2800 m altitude and 450 kra/hr indicated speed.
Let us define the radius of the first turn for a mean altitude of 2000 m,
keeping in mind that the indicated speed of 450 km/hr corresponds to a mean
air speed of 486 km/hr (135 ra/sec) :
Vcor 135= 18 200
= 6900 m
glg-i 9.81-0.268 9.81-0.268
Where y = 20° (tan 20° = 0.363), we produce r = 5100 m.
Let us determine the radius of the third turn when flying at V. ,
•^ " md
= 350 km/hr and y = 15°:
Note: Tg =. Tan
9480 _ ocnn ^
^= Si; 3600 m
At angle y = 20° and the same speed, the radius of the turn will be
On the fourth turn at V. , = 320 km/hr and y = 15° (landing gear down,
flaps down 15°), r = 3000 m, and at 20° bank, r = 2200 m.
Let us determine the time for a turn with a bank angle of 15°. An
increase in the radius of a turn also results in an increase in time required
to perform the turn. The formula presented for t is used to calculate
^ ^ cor
the time for a complete cornering maneuver, i.e., a 360-degree turn.
Usually, the aircraft performs turns of 180, 90 or fewer degrees.
The time required for a 180-degree turn (first and second turns performed
l!=0.64- ^^-0.5=161.5 sec==2 min 41.5 sec.
The time for the third turn is /ISO
^^0.64--^--^. 0.25 = 58 se.c.
The time for the fourth turn is
^ = °-^^-S-°-2^ = ^3--
The angular velocity of rotation during the performance of the fourth
^^il = -^ = 03rad/sec=l,7 deg/sec-
STABILITY AND CONTROLAB I LI TY OF AIRCRAFT
§1. General Concepts on Aircraft Equilibrium
In studying the stability and controllability of an aircraft, it is
represented as a body moving under the influence of external forces and
rotating under the influence of the moments of these forces.
In any flight, equilibrium of forces and moments acting on the aircraft
must be observed.
Equilibrium of the aircraft in flight is what we call the state in which
the forces and moments acting on the aircraft cause no rotation, i.e., the
given state is not disrupted.
In all flight modes, the aircraft should be balanced both in the
longitudinal and lateral directions. Balancing means achievement of equi-
libribrium of moments using the control surfaces in any flight mode.
Equilibrium of forces and moments acting on the aircraft is analyzed
relative to the three coordinate axes passing through its center of gravity.
The coordinate axes used (Figure 115) are the longitudinal axis of the
aircraft ox, the transverse axis oz and the vertical axis oy.
Figure 115 also shows the following moments: M is the yaw or track
angle, rotating the aircraft about axis oy, and is considered positive if the
aircraft rotates its bow to the left; M is the bank moment or the transverse
moment, rotating aircraft around the ox axis, and is considered positive if
the aircraft rotates toward the right wing; M is the pitch moment or the
longitudinal moment, rotating the aircraft about the oz axis, and is called
positive if the aircraft tends to lift its bow.
Equilibrium of the aircraft about these axes is divided into longitud-
inal equilibrium (about the axis oz) , transverse equilibrium (about the
axis ox) and track equilibrium (about the axis oy) .
Three characteristic forms of body equilibrium are known: stable,
unstable and neutral equilibrium. An example illustrating these forms of
equilibrium might be the behavior of a ball on surfaces of various forms. The
behavior of a ball on a concave curved surface characterizes stable
equilibrium, on a convex surface -- unstable equilibrium and on a flat
surface -- neutral equilibrium.
P = P - P
r r right r left
P >OifP .,^>P ,^
r r right r left
el' el e^
Figure 115. System of Aircraft Axes and Symbols Used for
Moments of Angular Velocities, Deflection of Control
Surfaces and Forces on Command Levers
Although aircraft equilibrium is a more complex phenomenon than the
equilibrium of a ball, in flight an aircraft may be in the stable, unstable
or neutral states. In correspondence with these forms o£ equilibrium, the
aircraft is called stable, unstable or neutral. An unstable or neutral
aircraft cannot satisfy the requirements of normal control in flight.
Static and Dynamic Stability
The stability of an aircraft is its ability to retain its flight regime
or return to its initial balanced regime in case of an arbitrary deviation
resulting from external perturbations, without the aid of the pilot.
At the present time, books on aerodynamics frequently divide stability
arbitrarily into static and dynamic stability, although in actuality an air-
craft simply has stability, meaning the ability of the aircraft to return to
movement at the initial kinematic parameters (velocity, angle of attack, etc.)
after a perturbation is removed or, as they say, the ability of the aircraft
to retain the initial flight regime.
Therefore, the stability of an aircraft consists of static stability and
good damping properties, which determine and characterize the quality of the
transient process when the equilibrium of the airccraft is disrupted. This is
frequently called dynamic stability.
Let us analyze these properties of an aircraft individually in somewhat
In flight, an aircraft is subject to the effects of turbulence of the
atmosphere, as well as short duration perturbations created by random devi-
ations of the control surfaces by the pilot, etc. The perturbing moments
disrupt the equilibrium of forces, causing the trajectory of the aircraft to
curve and the velocity of the aircraft to change. The summary movement of the
aircraft produced by adding the initial unperturbed and supplementary motions,
is called the perturbed movement.
Static stability means the property of an aircraft causing it to create
stabilizing moments when equilibrium is disrupted. For example, if a negative
pitching moment arises and acts on the aircraft when the angle of attack is
increased, this will be a stabilizing moment. Also, on the right wing
causes a moment to arise tending to turn the aircraft to the right, it will
also be a stabilizing moment.
Thus, if when equilibrium is disrupted, moments arise tending to restore
the initial equilibrium position of the aircraft, the aircraft is called
statically stable. The presence of static stability makes it possible for the
pilot to control the aircraft normally, and to take proper control actions in
Dynamic stability means the tendency of an aircraft, after a perturbing
force is removed, to restore the initial flight regime (velocity, altitude,
overloading, flight direction) without interference from the pilot. Dynamic
stability of the aircraft is characterized by: the period of damping of
oscillations T, the time of damping of oscillations T, (during which time the
initial amplitude of oscillations is decreased by a factor of 20) , the
decrease in oscillating amplitude A in one period m, = A /A (Figure 116) and
the relative oscillation damping coefficient c,. Coefficient c, determines the
quality of the transient process or, in other words, the intensity of damping
of oscillations from a perturbing movement.
In a dynamically stable aircraft, perturbed movement must be damped. The /183
movement may be either aperiodic (nonoscillating) , in which a perturbed
movement is rapidly damped, or periodic (oscillating), in which damping occurs
with a certain amplitude and requires somewhat more time (Figure 117) .
A neutral aircraft shows no tendency toward damping or increase in
perturbations (Figure 117 b) , while a dynamically unstable aircraft shows a
tendency toward increased amplitude of perturbations with time (Figure 117 c) .
Weak damping and oscillating periods which are too long are character-
istics of poor aircraft stability. As the period is increased, the perturbed
movement of the aircraft is "stretched out," i.e., extends over a longer
period of time.
Figure 116. Determin-
ation of Characteristics
of Short Period Damping
(Ai , A2 are amplitudes)
As we can see from Figure 118, the
behavior of a dynamically unstable aircraft
is characterizye by an aperiodic increase in
the pitch angle, that of a dynamically
stable aircraft by damping oscillations.
If neither stabilizing nor destabil-
izing moments arise when the aircraft
deviates from the equilibrium state, the
aircraft is called statically neutral
(Figure 118 c) .
Static stability alone is insufficient
to insure that the aircraft will have
dynamic stability. This requires additional
damping and inertial properties, as well as
a proper relationship of characteristics of static stability relative to the
various axes .
moments formed when
the aircraft is
rotated have a
tremendous role to
play in suppression
of oscillations and
provision of good
(pitch damping) is
created primarily by
the horizontal tail
surfaces, while yaw
damping) is produced
by the vertical tail
surfaces of the
aircraft . When
rotation about the
ox axis occurs, the
Figure 117. Characteristics of Perturbed Move-
ment of Stable (a), Neutral (b) and Unstable (c)
Aircraft (arrow shows initial equilibrium
pos i t ion)
wings create a transverse damping moment.
With weak damping, aircraft oscillations will be attenuated slowly,
particularly at altitudes of 10,000-11,000 m, and a great deal of time will be
required for restoration of equilibrium. With too strong damping, the return
to the equilibrium state is also delayed.
The inertial properties of an aircraft are characterized by its ability
to retain the state of equilibrium or its previous angular rotational
■■ II III I
velocity. The greater the moment o£ inertia, the more slowly the aircraft
reacts to deflections of the stick and pedals. Jet aircraft have high moments
of inertia relative to the y and z axes, since they have a relatively long
fuselage, in which the main mass of the load is concentrated about the center
of gravity. The moment of inertia relative to the x axis is less, since the
wing span is less than the length of the fuselage.
/// fff -— fff
wind gust wind gust wind gust
Figure Il8. Behavior of Dynamically Unstable (a),
Stable (b) and Neutral (c) Aircraft During Perturbed
§3. Controllability of an Aircraft
The controllability of an aircraft is an important piloting character-
istic, and means its capability to respond to the pilot's movements of the
rudder and ailerons with corresponding movements in space or, as they say, the /185
ability of the aircraft to "follow the control surfaces." In controlling the
aircraft, the pilot moves the stick and pedals and evaluates the behavior of
the aircraft by the forces on the control surfaces. By moving the various
surfaces, the pilot overcomes the inertial, damping and restoring moments
acting on the aircraft.
If the forces are extremely high, the pilot will become fatigued during
maneuvering. Such aircraft are described as being heavy to control.
Unnecessarily light control should also be avoided, since it makes precise
control of movements of control surfaces difficult and may cause the aircraft
The control surfaces should make it possible to balance the aircraft in
all flight regimes used. This is evaluated using balancing curves, which
characterize the change in balance angles of control surface deflection (and
correspondingly the position of the control levers, as well as the forces on
them) at various stable flight regimes as a function of a change in one of the
parameters determining the regime (for example, flight speed, M number, angle
of attack or slip angle, etc.).
The pilot also judges the controllability of an aircraft from the reac-
tion of the aircraft to deflections of the control levers during maneuvering.
Controllability is divided into three forms: longitudinal, directional and
transverse. The ability of the aircraft to rotate about the ox axis under the
influence of the ailerons is called transverse controllability, about the oy
axis under the influence of the rudder is called directional controllability
and about the oz axis under the influence of the elevator is called longitud-
Characteristics of longitudinal controllability include the amount of
elevator and stick travel required to change the aircraft velocity by a fixed
amount, as well as the force applied to the stick by the pilot. One of the
most important characteristics is the force gradient with respect to over-
loading AP /An , showing the force which must be applied to the stick to
change overloading by one unit.
The following parameters are used as characteristics of transverse
1) The force which must be applied to the stick to give the aircraft an
angular rotation velocity about the ox axis of 1 rad/sec:
where AP is the force applied to the aileron control lever;
Au is the change in angular velocity of 1 rad/sec;
2) The force which must be applied to the control lever to /186
balance the aircraft in straight line flight with a slip of one degree or a
bank of one degree:
pU-^ or P^^^^,
a A^ a A-r
where Ag is the change in slip angle of one degree;
Ay is the change in bank angle of one degree;
3) The change in angular velocity of a bank when the deflection of the
ailerons is changed by one degree:
where Aw is the ehange in angular velocity of the bank;
A6 is the change aileron angle of one degree.
The characteristics o£ directional controllability are the following
1) The force which must be applied to the pedals to impart an angular
velocity of 1 rad/sec to the aircraft:
where Ap is the force applied to the pedals;
Ao) is the change in angular velocity of 1 rad/sec;
2) the force which must be applied to the pedals to deflect the rudder
when the aircraft is balanced in straight line flight with a slip of one
degree or a bank of one degree;
^Pn „_ r.T APn .
P^ = iLCn or P^.
Ap Xi A-i
3) the change in angular velocity when the rudder is deflected by one
degree, i.e., the bank reaction of the aircraft to deflection of the rudder:
(Or = i
where A6 is the change in the rudder angle of one degree.
We can see from the definitions of aircraft stability and controllability
that they characterize opposite properties of the aircraft: stability must be
present to maintain the flight regime unchanged, while controllability must be
present to allow it to be changed. However, there is a certain interrelation-
ship between stability and controllability.
On a stable aircraft, the nature of the movements of the control levers
and required deflections during piloting are simplified, and it is easier to
determine the flight regime. It has been theoretically proven and confirmed
by practice that the higher the stability of the aircraft, the less the delay
and greater the accuracy with which it follows a deflection of the control
surfaces. Therefore, stability and controllability provide for complete /187
utilization of the maneuvering capacity of the aircraft, assuring the required
accuracy and simplicity of piloting and are an important condition for flight
§4. Centering of the Aircraft and Mean Aerodynamic Chord
The position o£ the center o£ gravity o£ an aircraft relative to the
wings is called the centering of the aircraft and is determined by the
distance (in percent) from the origin of the mean aerodynamic cord
(Figure 119) :
where b is the mean aerodynamic cord of the wing;
X is the horizontal distance from the lead point of the mac to the
center of gravity;
y is the vertical distance from the mac to the e.g.
Figure 119- Diagram for Determining MAC of
Trapezoidal Swept Wing (r . 1 .f ,=reference line of
aircraft; A, position of center of gravity
corresponding to tipping of aircraft onto tail)
Since y is small in magnitude, x is of primary significance in an
analysis of stability and controllability.
The center of gravity may be either above or below the reference line of
the aircraft, depending on the actual weight of the aircraft (fuel load) and
placement of motors.
In flight, the e.g. of the aircraft should be in strictly defined
positions in reference to the mac, guaranteeing continued stability and
controllability as the fuel is consumed. The fuel represents 25-45% of the
weight o£ the aircraft. In order to achieve the least displacement of the
e.g. in flight, the fuel is consiomed in a predetermined order, controlled by
an automatic device (Figure 120) .
As we can see from the graph, in order to remain within the required
range of centerings (x = 21-30% MAC), the loaded aircraft without fuel must
have X = 23.3-28.5% MAC (corresponding to sector AB on the figure). Then,
with any fuel load centering, of the aircraft will not go beyond these limits.
For example, if a centering of 26% mac was produced for the loaded aircraft
without fuel (landing gear down), when 8.5 t of fuel is taken on x = 26.7%,
or with 10.5 t — 24.3% MAC. After the landing gear are retracted, the
centering moves aft one percent and will amount to 26.7 and 25.2%
respectively. With a fuel remainder of 6.65 t, the centering will be furthest
to the rear, and with a remainder of 3.15 t -- furthest to the front.
With centering x = 42-50% MAC , for aircraft with motors on the wings
and 48-53% if the motor is located in the rear portion of the fuselage, the
center of gravity is located in the plane of the main landing gear struts;
with centering further to the rear, the aircraft may tip onto its tail
(Figure 119) .
3o%cA)i_^^c ( 1 . g.up4
7" 1 . g . dovyn
11 10,3 10
7 S,S5 e
Figure 120. Change in Centering of Aircraft in Flight As
a Function of Quantity of Fuel in Tanks (y. = 0.8 g/cm^)
§5. Aerodynamic Center of Wing and Aircraft. Neutral Centering
We know that there is a point on the cord of the wing about which the
moment of aerodynamic forces does not change when the angle of attack is
changed. For example (Figure 121) with an angle of attack a^ , lifting force
Y creates a longitudinal moment M relative to a certain point F
(Figure 121 a) . As the angle of attack is changed to a„, the lifting force
increases, but its arm length relative to point F is decreased as a result of
displacement of the center of pressure (Figure 121 b) . The new moment may be
greater than or less than the preceding moment. This depends on the way in
which the relationship between the values of force and arm length change. It
is possible to select a point F such that the value of the arm length changes
in inverse proportion to the aerodynamic force. Then, the moment relative to
this point will not change as the angle of attack is changed. This point is
called the aerodynamic center of the wing. Thus, if a.
> a_ > a^ and
L^ > t^ > 1-7, then Y^Z,
Y_Z_ is the constant moment of aerodynamic
force relative to the aerodynamic center of the wing with various angles of
attack. With wing shapes used, the aerodynamic center is located 23 to 25% of
the distance along its cord.
Figure 121. Explanation of Aerodynamic Center of Wing (a, b, c)
and of Ai rcraft (d)
We can draw an important conclusion from the definition of the aero-
dynamic center: the increments of aerodynamic forces arising when the angle
of attack is changed are applied to the aerodynamic center. Actually, force
Y = Y^ + AY, applied at cp„, can be divided into force Y^ applied to cp, and
force Y, applied at the aerodynamic center (Figure 121 b) .
Since the moment of force AY relative to point F is equal to zero, the
longitudinal moment of the wing at angle of attack a will be the same as at
angle of attack a .
The horizontal tail surfaces, like the wing, have their own aerodynamic
When the angle of attack is changed, additional lifting force arises on
the wing, and ends on the horizontal tail surfaces, applied to the aero-
dynamic centers of the wing and horizontal tail surfaces (Figure 121 d) . The
resultant of parallel forces AY and ■^Y, is applied at distances inversely
proportional to the values of these forces. The point of application of this
resultant is called the aerodynamic center of the aircraft. We must note here
that for aircraft of known types, both the horizontal tail surface lifting
force and its increment AY, are directed downward, no matter what the angle
of attack of the wing.
As we can see from the
figure, the moment of
supplementary forces relative
to the aircraft aerodynamic
center is zero; consequently,
the longitudinal moment of the
aircraft relative to this
center does not change when the
angle of attack is changed.
Therefore, the position of the
aircraft aerodynamic center
does not change when the angle
of attack is changed.
The aerodynamic center of
the aircraft is shifted to the
rear under the influence of
aerodynamic force increments
arising in the stabilizer,
fuselage and engine cells. For
example, if for the wing
without the horizontal tail
surface) x„ = 20-22% mac, for
Stabi 1 i ty Reserve
42 0,3 0,'f 0,5 0,6 0,7 0,8 M
Figure 122. Neutral Centering of Air-
craft with Respect to Overloads As a
Function of M Number (example):
a, Maximal indicated speed limita-
tion; b, Minimum permissible
indicated speed limitation
the aircraft x„ = 46-50% mac.
If the loads on the aircraft are so distributed that the center of
gravity of the aircraft corresponds with its aerodynamic center, the aircraft
becomes neutral in the longitudinal respect. In this case, the centering is
called neutral. Since in this case the longitudinal moment of the aircraft
will not change as a function of angle of attack, we must conclude that
neutral centering is the aerodynamic center of the entire aircraft^. Neutral
aircraft centerings are calculated for various altitudes and flight speeds
(Figure 122) .
^ T. V. Ostoslavskiy , Aerodincxmika Samoleta [Aerodynamics of the Aircraft],
Oborongiz. Press, 1957-
As we can see from the figure, at Mach numbers M <_ 0.6, neutral centering
moves somewhat (by 1.1-1.7% mac) forward (relative to its initial values of
45-43% mac), while at altitudes over 6,000 m it shifts noticeably to the rear
as a result of the effect of the compressibility of the air.
For H = 11,000 m, the change in neutral centering from 42 to 49% mac
noted is explained by a displacement of the center of pressure of the wing to
the rear at M numbers greater than the critical M number of the wing profile
(approximately M > 0.7-0.72).
After determining the farthest forward position of the neutral centering,
the limiting rearward centering for operation is defined 10-12% less than
neutral centering. The distance between the neutral and limiting rear
centering is called the reserve of stability for centering.
§6. Longitudinal Equilibrium
Figure 123- Diagram of Forces and Moments Acting
on Aircraft About Transverse Axis
The pilot maintains longitudinal equilibrium or balancing by using the
elevator and selecting the necessary motor thrust. Any stable flight regime
is characterized by angle of attack a, flight speed V, altitude H and the
angle of trajectory inclination 0. In order to achieve longitudinal equi-
librium of the aircraft, the forces acting in the directions of the ox and
oy axes and the moments of these forces acting relative to the oz axis must be
in equilibrium (Figure 123) .
In horizontal flight, three conditions of equilibrium must be observed.
The first condition is: the lifting force of the aircraft Y must be equal
to its weight.
We know that the lifting force of an aircraft is created by the wing,
horizontal tail surface and partially by the engine nacelles. The lifting
force created by this fuselage is relatively slight, and is considered to be
part of the lifting force of the wing. As we can see from the figure, these
forces create moments about the transverse axis which decrease or increase the
angle of attack. The lifting force of the wing in cruising flight creates
negative pitch moment M = Y^-
The lifting force of the horizontal tail surface is directed downward,
and in all flight regimes used in practice creates the pitch moment
^tit = ^ht^t
In order for force Y, to be negative, the angle of attack of the
horizontal tail surface a, must also be negative.
As we can see from Figure 124, a, < a by the angle of downwash of the
stream e, fthe downwash of the stream results from the action of the aircraft
wing on the air stream) . Also, a, is influenced by the angle of the
stabilizer ^ (generally zero to -4°). Thus, a
/ stabi 1 ized chord
I 1 '-ci'r ^iZErEZtzS^S*-
Figure 124. Determination of Angle of Attack of
Horizontal Tail Surface ( rZ equals reference
line of aircraft; V equals flight speed; V^ equals
velocity of diverted stream)
For ordinary aircraft with the stabilizer on the fuselage at a flight
speed of M = 0.75-0.85 and c = 0.3-0,4, e = 2-3°. For example, with a = 3°,
e = 2.68° and (f> = -2°, angle a^^ = 3° - 2° - 2.68° = - 1.68°. The greater the
angle of attack (greater the lifting capacity of the wing) , the greater the
downwash angle of the air stream.
In order to determine the summary longitudinal moment acting on the
aircraft, we must add the longitudinal moment resulting from engine thrust
fM ") to the moments o£ the wings and horizontal tail surface.
^ zen'^ ^
The axis of an engine located in the rear portion of the fuselage is
placed above the center of gravity of the aircraft; therefore, the thrust of
the motors creates a diving moment M = P Z .
" zen en en
Thus, the summary longitudinal moment acting on the aircraft is deter-
mined by the sum of the longitudinal moments of the wing, horizontal tail
surface and motor thrust.
Equality of the longitudinal moment to zero is the second condition of
The third condition for longitudinal equilibrium of an aircraft is
equilibrium of the forces acting in the direction of the ox axis. In order
for this condition to be fulfilled, the thrust of the engines must be equal to
the drag of the aircraft: P = Q.
If this condition is not fulfilled, the movement of the aircraft will be
accelerated or decelerated and, consequently, the lifting force will be
changed and the flight trajectory will curve.
These three conditions for longitudinal balancing of the aircraft are
fulfilled by varying the position of the elevator by the required angle and by
adjusting engine thrust, depending on velocity, altitude, flying weight,
centering, etc. We note that when equilibrium conditions are fulfilled, the
resultant of the aerodynamic forces and the thrust of the engines can be
considered to be applied to the center of gravity of the aircraft, and all
forces are balanced, i.e., P = Q and Y = G. Therefore, these forces will
not be shown on figures in the following, only the additional forces and
moments and their increments arising under the influence of perturbations
§7. Static Longitudinal Overload Stability
A disruption in longitudinal stability of an aircraft is accompanied by a
change in the angle of attack at flight speed, the angle of attack changing at
first more rapidly than velocity. Subsequently, on the other hand, the speed
changes more rapidly. For example, by pulling the stick toward himself
quickly, the pilot can increase the angle of attack by a factor of two or
three times or more. However, in order for the aircraft to change its flight
speed by 1.5 times, he must use not a fraction of a second, but dozens of
seconds or even several minutes. This sharp difference in the nature of the
change in angle of attack and velocity when longitudinal equilibrium is
disrupted has made it necessary to distinguish between longitudinal angle of
attack stability (overload stability] and velocity stability.
The stability of the aircraft in the first moment after equilibrium is
disrupted is characterized by its angle of attack stability or overload
stability. This name is given to this form o£ stability since when the angle
of attack is increased or decreased (at constant velocity) the lifting force
is changed, so that the overload is also changed.
The value of the overload shows the extent to which the external load is
greater than the weight of the aircraft. The overload is always related to
the direction in which it is being analyzed. In flight, the external loads
acting on the ox and oz axes are slight. Thus, the drag of the aircraft,
which is 10-12 times less than the weight of the aircraft, acts along the ox
axis; the loads arising only during slipping or as a result of side wind gusts
act along the oz axis.
Figure 125. Forces Acting on Aircraft Entering a
Vertical Wind Gust
Therefore, the main overload is that acting in the direction of the oy
axis. In this case, the external load is the lift of the aircraft Y and
If constant c is maintained at the given aircraft speed, the lifting force
will also be constant. The overload will also be unchanged, equal to zero.
An aircraft is called overload stable if it tends to retain the overload
of the initial flight regime independently, without interference by the pilot.
If an aircraft is overload stable, when the angle of attack is changed
the moments change so that the rotation of the aircraft which they cause
results in disappearance of the additional overload. Let us assume that an
aircraft in straight and level flight with an overload n = 1 and velocity V
enters an ascending current with velocity W (Figure 125) . This causes the
direction of the resulting velocity to be changed, causing an increase in the
angle of attack and an increase in lifting force AY (always at the aerodynamic
center) or an increase in overload An = AY/G. If force AY causes a diving
rotation of the aircraft, the aircraft is stable. As we can see from the /195
figure, this will result if the center of gravity of the aircraft is located
in front of the aerodynamic center. Consequently, the appearance of a diving
moment when the angle of attack is increased is a characteristic of overload
stability of the aircraft.
If the external action led to a decrease in the angle of attack, a
pitching moment would arise which would tend to increase the angle of attack,
i.e., restore the initial overload regime.
With a certain position of the center of gravity (at the aerodynamic
center) , the aircraft will not react to disruption of equilibrium and will
show no tendency either to return to initial overload or to further movement
away from the initial value. This position of the center of gravity, as was
discussed above, is called neutral centering. Movement of the center of
gravity to the rear, behind neutral centering, results in the appearance of
overload instability of the aircraft, since force AY will cause an increase in
the pitch moment arising when equilibrium is disinpted.
Thus, overload stability of the aircraft will be characterized by the
position of the center of gravity of the aircraft relative to the neutral
centering or the aerodynamic center. Therefore, in addition to leading
centering, which defines the capability of balancing of the aircraft in
flight and during landing with maximum displacement of the elevator, we also
determine permissible rear centering from the condition of provision of normal
overload stability for the aircraft.
We can see from our analysis that a change in overload stability in
flight may result from a change in the position of the center of gravity, as
well as a change in neutral centering -- the aerodynamic center of the
aircraft. The neutral centering of the aircraft may change in flight as the
velocity or engine operating mode is changed, as well as when control is
released. If overload stability increases with unchanged center of gravity,
this indicates an increase in the distance between the center of gravity and
neutral centering. On the other hand, if overload stability decreases, the
distance between the center of gravity and neutral centering must be
As a rule, neutral centerings are determined for aircraft with fixed
elevator; if the control is released, centering is moved forward by approx-
imately 1-2% mac.
The operating mode of the engine influences the longitudinal stability of
the aircraft to overloads. In jet aircraft, the downwash of the air stream in
the area of the stabilizer changes not only under the influence of the wing,
but also due to the effect of the exhaust gases of the jet engine on the
surrounding medium. The stream leaving the engine at high velocity attracts a
certain amount of the surrounding air along with it. This surrounding air
changes the direction of the stream as it approaches it. Usually, the
horizontal tail surface is located above the stream (Figure 126) , and the
resultant of the air flow toward the stream decreases the angle of attack of
the horizontal tail surface (makes the stream downwash more negative) .
During a climb, the operating regime of the engines is nominal and the
stream leaving the motor is at its highest power level. The downwash of this
stream is then maximal and decreases the angle of attack of the horizontal
tail surface significantly (makes the angle of attack a, considerably-
When the angle of attack of the wing is increased (aircraft enters a
vertical wind gust) the angle of attack of the* horizontal tail surface becomes
more negative due to the increased downwash of the stream resulting from the
change in lift of the wing and also from the stream of gases. The resultant
of the increase in lifting force of the horizontal tail surface ^Y, , applied
at its aerodynamic center and directed downward, will decrease the restoring
moment of the horizontal tail surface and make the aircraft less effective in
returning to its initial flight regime. This indicates the decrease in
longitudinal stability reserve, i.e., the aerodynamic center of the aircraft
is moved forward along the cord as a result of the engines operating at
Figure 126. Pumping Effect of Jet Engine Exhaust
Gas Stream on Surrounding Air Stream
When gliding at low engine setting, the influence of the stream from the
engines can be ignored. In this case, the downwash of the stream on the
stabilizer will be determined by the influence of the wing alone. The angle
of attack of the horizontal tail surface increases (becomes less negative) and
its effectiveness is increased. Longitudinal overload stability of the
aircraft is increased. This increase in aircraft stability is equivalent to a
displacement of the neutral centering of the aircraft (aerodynamic center)
backward along the mac. This is why aircraft stability is slightly lower in a
climb than in a glide.
Overload stability of the aircraft can be estimated by the overload force
gradient AP , /An .
" el y
§8. Diagrams of Moments
The degree of longitudinal stability o£ an aircraft is determined by-
wind tunnel testing. Models are tested with various deflections of the
elevator, and the longitudinal moment M is measured using special scales. By
determining moment M at several sequential angles of attack, we can construct
graphs called moment diagrams ra = f (a) for various M numbers (Figure 127) .
m, A pitch
Figure 127. Coefficient of
Longitudinal Moment m As a
Function of Angle of Attack
(6 ^, = 0)
The longitudinal moment coefficient [a dimensionless quantity such as c
and c ) can be determined using the following formula:
The pitch moments may be either positive or negative.
Actually, in flight the elevator always has some balancing deflection.
The angle of attack at which m = 0(M = 0) is called balanced, since at this
angle a the aircraft is in the state of equilibrium. As we can see, as the
angle of attack is increased to a fC ) the aircraft acts stably, since
^ sup ^ y sup^
the diving moment which arises causes it to return to its initial position.
A random decrease in the angle of attack by -Aa causes a positive pitch
moment((+m ) which returns the aircraft to its initial equilibrium position,
corresponding to location of the center of gravity in front of the aero-
Sector AB of curve m = f(a) corresponds to insensible equilibrium of the
aircraft, since an increase in the angle of attack causes no change in the
longitudinal moment. Sector BC of the moment diagram corresponds to (over-
load) unstable behavior of the aircraft: when the angle of attack changes, an
additional positive pitch moment arises, tending to increase it still further.
§9- Static Longitudinal Velocity Stability
A velocity stable aircraft is one which restores its assigned velocity
without interference of the pilot after perturbation. For simplicity of
discussion, we can consider that the angle of attack does not change when the
velocity is changed. Let us assume that an aircraft flying horizontally at
constant velocity V begins to descend for some reason (Figure 128 a) . Due to
the descent, it increases its velocity by AV.
Figure 128. Behavior of Aircraft After Random
Descent (a) and Flight Trajectory of Velocity
Unstable Aircraft (b)
If angle of attack a or c remains unchanged, due to the increase in
velocity, the lifting force also increases by AY. Due to this, the total
lifting force becomes greater than the weight components and the aircraft
trajectory begins to curve upward, the velocity begins to decrease, and AY
also begins to decrease. After achieving its initial altitude (point c) the
aircraft will have its initial velocity V, but its trajectory will be curved
slightly upward. Therefore, the aircraft will continue to climb. Due to the
increase in altitude, the velocity will begin to decrease, i.e., AV will
become negative. This makes Ay negative, and the trajectory begins to curve
downward, etc. Thus, the aircraft will oscillate.
If the aircraft is velocity stable, these oscillations will be damped and
the aircraft will come out of oscillations at its initial altitude and
velocity. Oscillation damping occurs due to the fact that the forces involved
in the oscillating process are always directed so as to even the trajectory.
As we can see from the figure, when the trajectory is deflected downward and
AV is positive, positive increments AY are also produced; when the trajectory
deflects upward and AV is negative, negative AY results. Naturally, in
practice the pilot will not wait until the oscillations damp out of their own
accord. He takes control of the aircraft and immediately eliminates them.
However J it sometimes occurs that, in spite o£ an increase in velocity,
the lifting force is not increased, but rather decreased, since the lifting
force depends not only on velocity, but also on c . Due to the influence of
compressibility in flight at large M numbers or due to elastic deformations,
c may increase so sharply with increased velocity that the lifting force
decreases rather than increases. In this case, the flight trajectory will
curve ever more sharply downward (if the pilot does not take control of the
aircraft quickly using the elevator) , the speed will increase and the aircraft
will go into a dive (Figure 128 b) . No return to the initial position occurs.
Figure 129- Dependence of Force on Elevator
Control on M Number (nominal mode, horizontal
flight, H = 10,000 m, tremor deflected by t = 2.3°)
It is easiest for the pilot to judge velocity stability from the nature
of the cliange in forces on the control stick when the aircraft velocity or
M number changes. As we know, balancing of an aircraft at various speeds of
horizontal flight requires varying force on the stick.
Figure 129 shows the forces required to balance the aircraft at various
M numbers (see §10 of this chapter). Thus, where x = 28% mac and M = 0.62,
the force on the stick is equal to zero, since the aircraft is balanced by
the trimmer and, consequently, the stick can be released in this position.
This is the balanced regime. As the aircraft accelerates to large M numbers,
pressure forces will arise on the stick (if the trimmer is left in its initial
position), indicating that the aircraft is velocity stable. Actually,
suppose the M number increases to 0.74. We can see from the graph that in
order to hold this new speed (M = 0.74), the pilot must apply a pressure of
P = +10 kg to the stick, i.e., create a diving moment with the elevator in
order to balance the positive pitch which has arisen.
We can conclude from the above that if at M = 0.62 with the stick
released, a random increase in M number to 0.74 occurs, a positive pitch
moment should act on the aircraft, increasing the angle of attack, and the
aircraft will return without interference from the pilot to its initial
velocity (M = 0.62). Consequently, this aircraft is velocity stable. A
similar picture will occur if the velocity is decreased.
■ ■■■I I
At Mach numbers M > 0.8, the compressibility of air begins to have a
significant influence, and the pressure force resultant (center of pressure)
is displaced rearward; an additional negative pitch moment begins to act on
the aircraft. Therefore, whereas at M = 0.74, a force of 10 kg must be
applied to the stick, at M = 0.82 the force will only be 8 kg, i.e., the
pressure force on the stick is decreased, and some velocity instability
appears. However, since the aircraft wing is swept, the phenomenon of pulling
into a dive (during acceleration), a property of velocity instability, is not
A decrease in pushing force is observed in a narrow range of M numbers,
then beginning at M = 0.88-0.9, the force required increases once more,
indicating the appearance of a considerable positive pitch moment, increasing
with increasing M number.
§10. Longitudinal Controllability
Longitudinal overload stability determines the characteristics of
longitudinal controllability of an aircraft, related to rotation of the air-
craft about the oz axis and creation of overloads .
If the performance of a maneuver requires that the overload be changed,
the pilot should do this by deflecting the elevator, disrupting the equi-
librium and overcoming the moments attempting to return the aircraft to its
The primary moments preventing rotation of the aircraft about the oz axis
are: the aircraft overload stability moment, the damping moment and the
moment of inertia.
The greater these moments preventing rotation of the aircraft, the
greater the angle to which the elevator must be deflected and the greater the
force required at the control stick in order to change the overload. Since
the pilot feels the value of force applied to the stick and the overload
resulting from it, longitudinal controllability of the aircraft can best be
evaluated by the gradient of overload force AP , /An and the elevator travel
used A6 ,/An .
The overload force gradient is numerically equal to the ratio of /201
additional force Ap on the stick to the increase in overload An produced as
el y ^
a result of this force.
Let us assume that the aircraft is performing horizontal flight and
n = 1 (Figure 130). Then, in order to produce n = 2, the pilot must pull
the stick toward himself with a force of 40-70 kg (for small M numbers, 40 kg
and for M = 0.7-0.8, 50-70 kg). Since overload stability characterizes the
ability of the aircraft to retain the initial overload regime, obviously the
higher the stability the greater the force required at the control stick to
change the overload.
Overload Force Gradient
d Elevator Travel
a Function of M Number
H = 10,000 m)
PVe can also see on
Figure 130 that if the centering
moves further forward, the force
required to change n increases.
This is explained by an increase
in the distance between the
center of gravity of the aircraft
and its aerodynamic center.
Thus , the further forward the
centering of the aircraft, the
heavier it is to control.
The limit in forward
centering is selected from the
condition of aircraft balancing
during takeoff and landing.
In order to exclude (during
takeoff) stream separation from
the horizontal tail surface, the
elevator can be deflected 20-25°
upward. During landing, the
pilot should increase c to
c , J . By pulling the stick toward himself, he increases the angle of attack,
creating positive pitch moments. When the angle of attack is increased, an
increase in lift Ay occurs, applied to the aerodynamic center and creating a
negative pitch moment opposing the pilot. The greater the distance between
the aerodynamic center and the center of gravity, the greater this hindering
moment will be. Since the movement of the elevator is considerable at low
velocities, it may be found that the limitng deflection of the elevator is
insufficient to tilt the aircraft to its landing angle. Therefore, the
maximum rearward position of the center of gravity is fixed so that the
permissible deflection of the elevator is sufficient to allow the pilot to
The usage of an adjustable stabilizer makes it possible to fly in
aircraft with more forward centering, since in this case the effectiveness of
the elevator is increased.
Usually, some reserve in elevator deflection (3-4°, but no less than 10%
of the complete deflection of the elevator) is installed.
Let us now analyze the deflection of the elevator A6 -./An necessary to
create an additional unit of overload. As we can see from Figure 130, as the
velocity increases, the effectiveness of the elevators also increases sharply.
1 — 1—
Figure 131 • Balancing Curves of Elevator
Deflection (produced as a result of flying
tests): a, In straight flight at nominal eng i
operating mode; b, Coming in for a landing
For example, whereas
at M = 0.5, the
elevator must be
deflected by 8° in
order to cause a
double overload, at
M = 0.78 the
is only 4°.
curves, showing the
on M number, are
also used to char-
(Figure 131) .
these curves, for
= 28% mac) , maintenance of longitudinal equilibrium
the elevator be deflected from its neutral position
M = 0.74, 1.5° downward; at M = 0.82, the balancing
elevator is decreased slightly, becoming once again
at M = 0.62 requires that
by 1.2° downward; at
downward deflection of the
Thus, as the aircraft accelerates from M = 0.62 to M = 0. 74, longitud-
inal balancing requires that the elevator deflection be moved downward by
0.3°, while further acceleration to M = 0.82 requires that it be decreased by
the same amount.
Beginning at M = 0.88-0.9, the positive pitch moment increases sharply,
and the elevator must be deflected considerably downward.
§11. Construction of Balancing Curve for Deflection of Elevator
Using the moment diagrams for various deflections of the elevator, we can
determine for these deflections coefficients c_. with m = 0(c ,c ,...,c )
and construct the balancing diagram for deflection of elevator as a function
of c (Figure 132) . The left branch of the graph (left of c ) can be
produced by wind tunnel testing of a model, while the right branch can only
be produced in test flights testing the stability and controllability of the
aircraft at high angles of attack; in these tests, the deflection of the
elevator as a function of c is determined for each M number. For this, the
aircraft is placed in the regime c > c and held in this regime until the
beginning of "pickup," allowing us to determine the degree of stability of the
aircraft and sufficiency of the elevators to bring the aircraft out of this
regime. The aircraft is also braked in order to determine the minimum
velocity and nature of its behavior at this velocity.
The balancing curves on
Figure 133 give us an idea of the
nature of the dependence of elevator
deflection 6 , for aircraft equilibra-
tion with respect to longitudinal
moments at stable flight regimes on
coefficient c . As we see, these
curves are similar in form to the
moment diagram, for which proportion-
ality of the deflection of elevator to
the coefficient of longitudinal moment
m is also characteristic.
In order to record the deflec-
tions of the elevator during flight
tests, the aircraft is accelerated to
M = 0.65-0.85, and then
at constant M number, the elevator is
"fed" toward the pilot in order to
cause the aircraft to climb. This
"feeding" of the elevator is performed
with c with constant increase in overload n to 2-3.
y sup y
Let us analyze the movement of the aircraft upon transition to large
angles of attack (c > c ), when the pilot is controlling the aircraft.
° y y sup ^ *"
Let us assume that as a result of the influence of a powerful ascending
air current (or as a result of creation of an overload in a test flight) the
Figure 132. Construction of
Elevator Deflection Balancing
aircraft arrives at c
> c (Figure 133)
ypu ^ '=' ■'
It was noted in chapter II that
if c is exceeded, longitudinal stability of the aircraft may be dis-
y sup > ', y J
rupted, since as a result of redistribution of pressure on the wing, so-called
"capture" -- involuntary progressive increase in the angle of attack -- occurs.
The angle of attack near which "capture" occurs is called the "capture"
angle of attack (the coefficient c and overload above which "capture" begins
are named similarly) .
If at the moment of capture the pilot moves the elevator downward by
by the time the angle of attack a (c -, )
is achieved for which 6
. ,-^ e 1 max
(Tel niac considering deformation
— - M=0.75
^ ^Xt=30% mac / I ,
increase in the angle
of attack does not
occur and the aircraft
is balanced at angle of
attack a^ and will
retain this angle ^.
The behavior of an
aircraft in this curved
flight with n > 1 will
be characterized by a
tendency to increase
the pitch angle without
increasing the angle of
Figure 133- Required Elevator Deflection As
a Function of c
In order to return
the aircraft to its
initial flight regime,
the pilot still has the
elevator reserve A6 ,
separating the balancing elevator deflection from the maximal deflection,
corresponding to complete deflection downward (to the stop) . The further the
pilot moves the elevator downward from this balancing position, the greater
the angular velocity with which the aircraft will begin to decrease the angle
of attack, i.e., the more rapidly the overload will be decreased to unity.
A position should not arise in which the required downward elevator
deflection to restore balancing is greater than that available, including
consideration of deformation of force transmitting hardware. Otherwise, it
will be impossible to balance the aircraft, and the pilot will not be able to
return it to the initial flight regime.
Figure 133 shows that with more forward centering (25% mac) the elevator
reserve is greater, and the controllability is better. This results from the
fact that with forward centering in the initial balancing regime the elevator
control stick must be held closer to the pilot than with rearward centering
and, consequently, the elevator reserve to maximum deflection is increased.
It has been noted in the process of flight tests that after an aircraft
is put in a high overload position, soaring requires that a positive pitch
moment be created by applying a force of 80-100 kg to the stick. This force,
which equalizes the aerodynamic load acting on the deflected elevator, deforms
the force transmitting elements, shortening them. As a result, full forward
deflection of the stick did not result in full deflection of the elevator.
With maximum deflections of the elevator (29-31°) the actual angle of position
^ M. v. Rozenblat, Piloter o Peregvazke [To the Pilot Concerning Overloading],
Asroflot Redizdat Press, 196^.
was only 24-25°, due to deformation (Figure 134).
The only method of creating a reserve of elevator movement for aircraft
control in this case is unloading of the control cable by using the elevator
When the trimmer of the elevator is deflected, the hinge moments
decrease, and the deflection of the elevator is increased as a result of
unloading of the control cables.
During the process of flight tests of an aircraft at high angles of
attack, the following peculiarity was discovered. We know that when a back-
swept wing moves at high angles of attack, flow separation begins where the
ailerons are located. This leads to a change in the aileron hinge moment such
that both ailerons tend to move upward by approximately 2-4°. This phenomenon
has come to be called "floating" of the ailerons. In its effect, it is
equivalent to an additional deflection of the elevator upward, since it causes
an additional loss in lift at the terminal portion of the wing where the' lift
properties are worsened by the separation. "Floating" of ailerons worsens
longitudinal instability of the aircraft with swept wings at high angles of
attack and makes capture of the aircraft even sharper. The design-
aerodynamic measures analyzed in §3 of Chapter III improve the overload
stability characteristics of a swept wing aircraft at high angles of attack.
cable de format
Figure 13^. Diagram of Deformation of
Control Cable As Vertical Gust Is Being Over-
"Floating" of the
ailerons can be reduced
by fastening the
external portion of the
ailerons using special
mechanical devices or
by decreasing the size
of the ailerons. The
transition to hydraulic
control) installed in
the immediate area of
the ailerons can
deformation of the
control cable, which
of the ailerons.
A pilot flying a
passenger aircraft with a swept wing should avoid areas with strong
turbulence, in which the characteristics of longitudinal overload stability
appear so unfavorably.
§12. Vertical Gusts. Permissible M Number in Cruising Flight
During flight through atmospheric turbulence, in tensive and frequent
vertical gusts of air result in large longitudinal and lateral oscillations of
the aircraft. The accelerations arising in this case lead to the appearance
of inertial forces characterized by overloads on the aircraft. A vertical /207_
gust is a vertical air movement resulting in an increase in overload in not
over 2 sec.
The horizontal components of wind gusts have no essential significance
for the movement of the aircraft. For example, horizontal wind gusts up to
6-15 m/sec cause slight velocity pulsations in modern aircraft flying between
200 and 250 m/sec, and create slight overloads, whereas vertical wind gusts at
these speeds cause 10-15 times more overloading^.
Longitudinal overloading (or more accurately an increment in overloading)
acting in the horizontal plane can be determined according to the following
where Av is the change in velocity resulting from an oncoming gust;
At is the time of action of the gust.
Thus, if a horizontal wind gust causes a velocity variation of 11 m/sec in
two seconds, the increment to the longitudinal overload will be
An = 11/2-9.81 - 0.56: with a time of action of three seconds. An = 0.37.
The sign of the overload will depend on whether the gust is a headwind or
tailwind. In the case of a headwind gust, the sign will be plus (the crew and
passengers will be pressed against the backs of their seats) , and with a
tailwind gust the sign will be minus (the crew and passengers will be pulled
away from the backs of their seats).
What must the velocity of a vertical gust be in order for the aircraft to
be brought to c or to the mode of involuntary increase in overload
" y sup
("capture")? As we can see from Figure 135, at M = 0.8 when a gust of W. ,
an aircraft with an initial value of c , j- will reach c , while the effects
of a gust at W. will cause it to reach c . . In this case, the
^ 1 capt y capt
balancing position of the elevator will be insufficient to return the aircraft
to its initial parameters.
In order to estimate the effects of a vertical air stream on the wings of /208
an aircraft, we must use the so-called velocity of the effective gust. The
indicator effective gust W. ^ differs from the real indicator gust (measured
under concrete conditions) , since there are no sharply differentiated vertical
^Kul ik, M'. M.V Ohosnovaniye rekomendatsky po P-ilotirovcmiyu Samoletov -pri
Foletakh v Zonakh Atmospemoy Turhulentnosti [Basis for Recommendations'
for P(lQti,ng Aircraft on Flights in Zones of Atmospheric Turbulence]
GqsNII, GA Press, I963.
Figure 135. Determination of
Effective Indicator Vertical
Gust Bringing Aircraft to
c and c ^: 1, Initial
y sup y capt
balancing regime; 2, Effective
movements in the atmosphere, as a
result of the influence of viscosity
of the air. There is always a
transition zone, in which the rate of
the vertical component varies from
zero to some value W. _. Various
aircraft with their inherent specific
features of aerodynamics react differ-
ently to the same gust. For example,
it has been established that for
aircraft with swept wings, W. . =
= 1.11 W. . ^®^
Calculation of the velocity of an
effective vertical gust is performed
using the formula
where Aa is the increase in angle of attack calculated from a.
V. is the indicator velocity of the aircraft.
Let us assume that the pilot does not interfere in control and that the
elevator is "clamped" in the initial balanced position. Let us calculate the
gust speed W. j, required to bring the aircraft to c . The flight is
^ ^ lef n to y. 5^p 6
performed at c = 0.35 and a = 3° at M = 0.75 and H = 10,000 m. In this
case c = 0.715 and a = 7.2°. Let us determine: the increment of angle
y sup sup ^
of attack Aa = 4.2° or 0.073 rad, the indicator velocity V. = 475 km/hr =
= 132.0 m/sec, so W. ^ = 1.11 V.Aa = 1. 11 •132-0.073 = 10.7 m/sec.
' lef 1 '
The effective indicator vertical gust corresponding to the beginning of
involuntary increase in overload -- "capture" with fixed control -- is
calculated using the same formula, except that the increase in angle of attack
is selected from a, ^ to the beginning of "capture." Thus, for the same
1.11-132-0.157 = 23 m/sec.
When a vertical gust at 10.7 m/sec acts upon the aircraft, it goes to
c , while where W. ^ = 23 m/sec, the "capture" regime is begun, and a
self-sustaining increase in overload and vibration of the entire aircraft
As we can see from Figure 136, at M = 0.75, the reserve for a vertical
gust for the weight and flight altitudes here analyzed is maximal. At
HIS (118 0.8 H
Figure 136. Permissible
Vertical Gust As a Func-
tion of M Number of
Fl ight (TU-124 aircraft)
M = 0.75-0.78, a slight reduction is
observed, and at M > 0,78 this reserve is
somewhat greater. Therefore, for this
aircraft, the maximum permissible M number
in horizontal flight is 0.78, in order to
retain a sufficiently high reserve of
vertical gust stability.
§13. Permissible Overloads During a
In addition to vertical air gusts, an
aircraft may be subjected to the action of
extended ascending or descending air
currents, which cause considerable vertical
displacement of the aircraft, independent
of pilot action.
In stable horizontal flight, the sum
of vertical forces acting on the aircraft
is equal to zero and the overload
When the aircraft crosses a vertical gust, the angle of attack increases
rapidly and consequently the lifting force increases as well. All of this
causes vertical and angular displacement of the aircraft, which in turn once
more influences the angle of attack. In this case, the overload
_Yw _ Y+LY
The increment of overload An occurs as a result of the summary increment
of angle of attack resulting from the influence of the vertical gust and
angular displacement of the aircraft caused by the gust. The overload acting
on the aircraft can be represented in this case by the following expression:
1^ ^ G
(the "plus" sign relates to an ascending gust, the "minus" sign to a
where c is the tangent o£ the angle of inclination o£ curve c = f(a), i.e.,
the gradient o£ the change in coefficient c as a function of angle of attack
V. is the indicator velocity of the aircraft;
W. is the indicator velocity of the vertical gust;
K is a coefficient characterizing the increase in the vertical gust
(K = 0.85-0.95).
As we can see from the formula, the overload acting on the aircraft
depends on the flight speed and force of the vertical gust. Flights of high-
speed aircraft at high altitudes have shown that when the aircraft enters a /210
vertical gust with a certain velocity W. , the overload n (related to the
moment of action of the gust) is much less than n , but even in this case
^ ■' y max'
separation of the flow over the wing occurs, which may lead to rolling of the
aircraft. Usually, rolling is preceded by the appearance of a considerable
positive pitch moment, under the influence of which the aircraft climbs and
Therefore, limitations on overloads move along two lines: along the line
of aerodynamics, i.e., with respect to c , and along the line of strength
of the aircraft, i.e., with respect to the maximum coefficient of operational
In order to avoid exceeding c and prevent the aircraft from going
into a roll, permissible flight altitudes are established as a function of
flying weight (see Chapter VII, §8).
%]k. Behavior of Aircraft at Large Angles of Attack
At the present time, the separation characteristics, rolling and termin-
ation of rolling of aircraft with low stabilizers and engines installed on the
wings have been studied rather well.
However, there is still very little material available on the behavior of
aircraft with T-shaped tails and motors located in the rear portion of the
fuselage during flow separation at high angles of attack. The balancing
characteristic analyzed in §11 related completely to an aircraft with load
Let us analyze some features of the behavior of an aircraft moving into
large angles of attack. The flight speed of the aircraft corresponding to
c is called the minimum speed or the separation speed. The problem is
y max '^ r r r
that when c is achieved in flight, the flow separates, causing a sharp
decrease in the lift and a considerable increase in the drag. (The separation
speed for a smooth wing is represented as V , for the takeoff position of the
wing mechanism as V , for the landing position -- V .)
Due to the asymmetrical development of separation on the wings of the
aircraft, a banking moment arises and the aircraft rolls. By roll, we mean a
movement of the aircraft about the longitudinal axis such that the angular
velocity of rotation w > 0.1 rad/sec, i.e., greater than 6° per second.
In order to determine the minimum velocity corresponding to c , the
aircraft is decelerated at unit overload. Since the lifting force of the wing
depends on c V^, as the speed is reduced gradually, the value of c should
increase, which does occur, while the pilot, gradually pulling the stick
toward himself, shifts the aircraft into high angles of attack. The speed at
which sharp flow separation occurs is accompanied by rapid rolling of the
aircraft, and this is the minimum speed or the speed of separation V . A case
has been observed in which an aircraft developed such a high angular velocity /21 1
0) that it rotated by 180° in a few seconds.
With flaps down, the movement of the stick may not be sufficient to
achieve V or V . Then, the flight speed corresponding to maximum rearward
position of the stick is taken as the minimum speed.
As we can see from processing of strip chart recorders (Figure 137) when
an aircraft with a low stabilizer is decelerated at an altitude of 12,000 m
(flaps and landing gear up) after an indicated speed of 200 km/hr is achieved,
the aircraft maintains almost constant c =1.45 and overload n = 1 for
several seconds. The deflection of the elevator "upward" varies from 3 to
3.8°. At c = 1.5, a slight vibration of the ailerons and stick begins.
Rolling occurred at c =1.58 toward the right wing. In this case, the
angular banking velocity m reached 0.19 rad/sec (approximately 11 deg/sec) ,
and the nose dropped at 4 deg/sec. During the roll, the ailerons were
observed to move upward by 2-2.5° (negative deflection).
After 0.3-0.5 sec of roll, the pilot moved the stick away from himself
(6 , = +2°) and transferred the aircraft to lower values of c . In 3-4 sec,
el "^ y
the vibrations stopped. After the ailerons were moved to stop the bank, the
aircraft rapidly stopped rolling, the effectiveness of the ailerons being
sufficient. By pulling the stick toward himself (deflecting the elevator
"upward" by 2-3.5°), the pilot brought the aircraft back to horizontal flight
at 320-340 km/hr.
In order to determine permissible values of c , the elevator is "fed"
■^ y sup
at various values of M number (Figure 138). In order to improve safety, this
maneuver is performed at high altitude (about 12,000 m) . When the stick is
moved energetically backward, the aircraft is transferred to angles of attack
(high a ) at which "capture" or involuntary positive pitch occurs.
As we can see from the strip chart recordings, the aircraft first
accelerated, then when M = 0.66 was reached, the pilot began to increase the
overload by pulling the stick sharply back. The angular rate of rotation
about the transverse axis reached 12° per second (u = 0.2 rad/sec) . At this
point, the pilot slowed the rate at which he was pulling back the stick, and
the deflection was left constant at 3° "upward." The overload increased
sharply, reaching a maximum value of 2.8, and "capture" began at n = 2(c =
= 0.85) (sector ab) . As the overload increased to 2.05-2.2 (c -1), the
aircraft started vibrating and the ailerons began to "float" (deflection of
both ailerons upward due to elastic deformation of the control cable) . The
aircraft did not roll, but a bank did occur at 4-4.3 deg/sec. The maximum /213
"floating" of ailerons was 4.5-5°.
When the elevator was shifted at M = 0.7, vibration was noted at
c =0.85, while atM=0.8--atc =0.65. When the stick was moved for-
ward, the maximum balancing deflection of the elevator (M = 0.8 and c =0.9)
was 5.3°, and the maximum balancing force required to bring the aircraft back
to the initial regime was 60 kg.
It was noted in the process of testing that the warning vibration which / 214
arises as the minimum flight speed is approached is insufficiently intense to
be noticed by the pilot. A stronger vibration occurred at the moment of
"capture" or at the moment the aircraft started to roll.
In most aircraft as the separation regime is approached, the vibration of
the tail surfaces is noted due to interference between the tail and streams
from the wings of the aircraft. In those cases when vibration was not
observed, devices have been installed to cause artificial vibration of the
stick, warning the pilot that he was approaching the separation regime. From
the point of view of formation of vibration and rolling of the aircraft, it
is dangerous to perform a takeoff in which during the first stage of takeoff
the air speed is 20% higher than the separation speed V , as well as landing
during which the flight speed of the aircraft exceeds the separation speed
V by 30%.
H,n Vp , km/hr
^SOffO -400 ■-
_ j^lejft, thrust
Figure 137- Recording of Strip Chart Recorders
During Deceleration of Aircraft
Hm l^r '^'%r
Figure 138. Recording of Strip Chart Recorders As
Aircraft Is Transferred to n > 1
In horizontal flight (flaps up) at high altitudes when a zone of strong
turbulence is entered, separation may occur. In this case, if the aircraft
has satisfactory characteristics (a diving moment appears) and the pilot takes
control, the aircraft will eliminate the disruption of equilibrium.
The problem is somewhat worse as concerns the separation characteristics
of an aircraft with a high horizontal tail surface and motors in the tail
portion of the fuselage.
If in aircraft with low stabilizer, high slip angles e are created
immediately before separation, and the slipping of the stream disappears
immediately after separation, causing an increase in the angle of attack and
lifting force of the stabilizer (a, . - a ") , i.e., an increase in the diving
moment, in aircraft with T-shaped tail surfaces (high stabilizer) after the
stream separates from the wing, vortexes from the fuselage, and the stream
from the wing, engine nacelles and mounting struts strike the stabilizer,
causing a positive pitch moment (Figure 139) . This decreases the negative
significance of the longitudinal moment coefficient, and the aircraft has no
tendency to tip over on its nose. When the stabilizer is below the separated
stream zone, which occurs with very high angles of attack, the horizontal
tail surface creates considerable drag and a diving moment appears. In
connection with this, after separation, a positive pitch moment may arise,
making the situation worse; after separation begins, the elevator should be
fully deflected "downward." Therefore, in some aircraft with T-shaped tails,
a diving moment is created artificially using a "pusher" ("recoil" system)^.
This device, working from an angle of attack transducer located on the
fuselage, creates forces acting on the stick in the direction of a dive at an
angle of attack near a . This force should be high enough to overcome the
force applied by the pilot and should continue acting until the angle of
attack is decreased.
Figure 139- Flow Spectra Around Aircraft with
T-shaped Tail Surface After Flow Separation:
a, Angle of attack 3° greater than separation
angle; b. Angle of attack 18° greater than
separation angle; 1 , Ai r stream from wing;
2, Air stream from nacelles and struts of
^Zaxiibezhny'Tf AvidtTansTpoTt, No.' Vl, GosNlV GA Press, 1965.
In order to
prevent elimination of
overload by separ-
ation, the "pusher" is
equipped with a
special device with a
gyroscope which limits
the increase in angle
of attack as a func-
tion of the angular
velocity of the
beginning of separ-
The "pusher" can
also eliminate the
stable rolling mode
"long term positive
pitching moment) , in
which the aircraft
leaves the roll only
after a considerable
decrease in velocity
§15- Automatic Angle of Attack and Overload Device
The automatic angle of attack and overload device (AUAP) is used to warn
the pilot that the aircraft is flying at large angles of attack as the minimum
velocity is approached and during flights in bumpy air.
During flights using this device, the instantaneous angle of attack at
which the aircraft is flying and the vertical overload are determined. Also,
at each moment in time the value of the critical angle of attack is determined / 216
as a function of the M number of flight.
The device consists of a number of aggregates. The main units are:
1) the angle of attack measuring device, which measures the local angles of
attack in conjunction with the wind vane on the fuselage; 2) the critical
angle measuring device which outputs the required voltage as a function of the
M number of the flight; 3} the overload transducer, installed in the area of
the center of gravity of the aircraft; 4) an indicator device on the instru-
ment panel in front of the pilot. Using this device, the pilot can observe
the current angles of attack at which he is flying, the critical angle of
attack (more precisely, the angle of attack at which the automatic device
operates under the given conditions) and the vertical overload.
VJhen the aircraft enters a critical regime (the operating regime, which
is somewhat less than the permissible) the lower sector of the movable
critical angle of attack sector on this instrument corresponds with the arrow
indicating the instantaneous angle of attack (Figure 140). At this moment, a / 217
lamp with the inscription "ct^^' lights up in front of the copilot. Also, if
the aircraft undergoes overloads greater than those permissible, the arrow
indicating instantaneous overload approaches the sector of dangerous overloads
and the lamp with the inscription "n " lights up.
When either of these lamps lights up, the "attention" lamp on the display
begins to flash.
Adjustment of this device is performed individually for operation in
flight with all flaps and gear up and for flight with flaps down for takeoff
and for landing. For example, in the ordinary flying mode (flaps up), the a^^
warning lights up when angles of attack of 1.4-2° less than the permissible
angles are reached. These parameters are shown for one aircraft equipped with
the AUAP device in Table 13.
We can see from Figure 140 that the angle of attack reserve up to the
moment of operation is 1.8-3.2° (M = 0.7-0.82). For example, for M = 0.8, the
reserve from a, ^ = 3° to a = 5.2° is 2.2°, and the reserve to c is 4°.
hf op ' y sup
In order to achieve c = 0.7 in flight at M = 0.8, we must create an
y sup ° '
overload n = 0.7/0.275 = 2.52. However, at a = 5.2° (c = 0.53) , i.e. , at
y op ^ y ' ' '
overload n = 0.53/0.275 = 1.93, the "n " light comes on. The pilot's
y y sup ^
action in controlling the longitudinal attitude of the aircraft prevents the
aircraft from entering the dangerous rolling regime.
hf for H=10km'
Note: Commas represent decimal points.
0,1 0,2 0,3 0,t 0,5 0,6 0.7 0,8 M
The speed reserve from
the moment when the light
signalling the dangerous
regime lights up until the
minimum permissible speed is
reached is usually 25-40 km/hr
and the reserve before
rolling is 80-100 km/hr
With flaps down, the
automatic device also warns
the pilot in advance of any
deviation from the normal
regime. For example, where
a = 9-10° (near the angles
of attack used in landing and
takeoff) , transfer of the
aircraft into the nonpermis-
sible regime is signalled by
lighting of the "a " lamp,
Figure 1^*0. Operating Characteristics
of AUAP As a Function of M Number:
1, Movable sector of critical angles
o^ji 2, Sector of dangerous overloads;
3, Nonflashing lamp warning of danger-
ous n ; k. Flashing lamp; 5, Non-
flashing lamp signalling critical
§16. Lateral Stability
Lateral equilibrium of
the aircraft can be disrupted
by two factors which are
interrelated: slipping and
banking. Thus, if the cause
of a disruption of lateral
equilibrium is banking, as a
result of the force of
gravity an unbalanced lateral force will appear, applied at the center of
gravity, which will distort the trajectory of movement. The aircraft begins
to slip. In the same way, if the disruption of lateral equilibrium occurs as
a result of slipping of the aircraft, an increase in lateral force AZ occurs,
applied at the lateral aerodynamic center, the trajectory is curved and as a
result an unbalance transverse moment AM appears. The aircraft begins to
bank. Thus, when lateral equilibrium is disrupted, the aircraft begins
to rotate about the axes of ox and oy simultaneously.
The term lateral stability means the ability of an aircraft to return
to its initial position after any small perturbation independently,
without pilot action, except for unavoidable course deviation.
For a better understanding of lateral stability, it is methodologically
expedient to analyze first stability of the aircraft relative to the ox axis,
then separately relative to the oy axis. The former is called transverse
stability, the latter -- directional stability.
Simultaneous directional and transverse stability represent lateral
stability of the aircraft.
§17. Transverse Static Stability
Transverse stability is the ability of an aircraft to eliminate a
bank automatically, or, in other words, to bank in the direction opposite
to slippage. For example, if the aircraft slips to the right, the aircraft
should bank to the left.
In order for an aircraft to eliminate bank independently, it is
necessary that a transverse moment arise on the lower wing during slipping
such as to cause rotation toward the higher wing. The banking of the aircraft
itself has no direct influence on the magnitude of transverse moments. Its
influence is felt through slipping. The bank angle determines the slip angle
which is the direct cause of transverse moments.
The degree of transverse stability is evaluated according to the value of
transverse moment Am restored per one degree of slip angle 3, i.e., according
to the value of m , called the coefficient of transverse static stability:
In a transversely stable aircraft, when slipping occurs to the right wing /219
(positive slipping) , a negative transverse moment appears on the left wing,
and coefficient m is negative. The value of this coefficient is determined
primarily by the form of the wing and the height of the vertical control
surface. For swept wings with no transverse V, the transverse stability
coefficient is usually quite high, and must be decreased by giving the wing a
negative transverse V = -(1-3°). This decreases the moment of the bank
striving to bring the aircraft out of the slipping state.
Figure l4l. Change in Sweep Angle of Wing
During Slipping and Influence of Slipping on
Dependence of c on Angle of Attack
stability depends both
on the angle of attack
and on the flight
of the wing is also
quite important. The
increase in transverse
static stability with
c is explained as
follows. When a
swept wing slips, the
sweep angle of the
wing is changed
(Figure 141) . Where
the sweep angle is decreased (right wing) , the load bearing qualities
increase. The curve of the function c = f(a) for this wing is higher than
for the wing for which the sweep angle increases during tlie slip. We see from
the graph that at high angles of attack (more precisely at high values of c )
the difference in the values for the wings increases. Therefore, the higher
the angles of attack at which flight is performed, the greater the banking
moment created during slipping.
As a result, transverse stability of a swept wing is higher, the higher
the angle of attack. Whereas during climbing, horizontal flight and descent
(angles of attack 2.5-3.3°) the transverse static stability is within the
limits of normal values, during the landing regime it increases.
The increase in lateral static stability at high angles of attack has a
negative influence on the prelanding regime and may worsen the flying qual-
ities of an aircraft, causing it to rock and giving it poor damping char-
acteristics. Therefore, when the flaps are lowered (high values of c ) , when
flight is being performed at low speeds, the transverse static stability is
An increase in transverse stability of an aircraft at low angles of
attack is aided by aerodynamic deflection of the wings.
Aerodynamic baffles also extend the beginning of development of terminal
separation and help to increase the transverse stability of an aircraft at
high angles of attack.
§18. Directional Static Stability
Directional stability is the ability of an aircraft to eliminate slipping
automatically. During flight with slipping, as a result of lateral air
current against the fuselage, aerodynamic force Z arises, the moment of which
relative to the center of gravity creates a rotating moment M about vertical
axis oy. Normally, the point of application of the lateral force is behind
the center of gravity of the aircraft, as a result of which force Z tends to
rotate the aircraft (like a weather vane) toward the wing onto which the
aircraft is slipping. Quantitatively, the degree of directional stability is
determined by the value of stability coefficient m^. Physically, coefficient
m defines the amount of increase in rotational moment M when the slipping
angle 6 changes by one degree, i.e.,
/7Ivf = •
The greater ra , the greater the directional stability of the aircraft and the
more intensively it eliminates slipping.
Modern aircraft have sufficient directional stability, coefficient m is
negative, i.e., when the aircraft slips over onto the right wing (positive g)
a directional moment appears to rotate the aircraft to the left.
Directional stability of aircraft is provided primarily by the vertical
119. Lateral Dynamic Stability
Let us assume that an aircraft is banked onto the right wing under the
influence of external perturbation. This results in right slippage, and the
trajectory of the aircraft is bent to the right. Further movement of the
aircraft depends on the ratio between transverse and directional stability.
Let us assume that the transverse stability is greater than the directional
stability, i.e., m is greater than m . In this case the bank is rapidly
eliminated, the aircraft moves from right bank to left bank and begins to slip
on the left wing. However, since the slipping is not completely eliminated,
once more a banking moment onto the right wing appears. The aircraft goes
into a right bank once more. Thus, a rocking of the aircraft occurs, called
lateral oscillating instability.
On the other hand, if m is less than m , i.e., the directional moment is
greater than the transverse moment, after the aircraft is banked, the bank is
retained, but the slipping is rapidly eliminated. The remaining bank curves
the trajectory, i.e., the aircraft descends in a spiral to the right. This is
known as lateral spiral instability.
The dynamics of the lateral movement of the aircraft under the influence
of external conditions and its behavior iinder the influence of the pilot's
actions are determined in these examples not only by the sign and magnitude of
coefficients m and m but also by the presence of certain relationships
between them. Therefore, the magnitude of k, which is directly dependent on
the ratio m^/m and numerically equal to the ratio of angular velocities of
bank and yawing, is very important in lateral dynamic stability as well as the
controllability of the aircraft.
This parameter characterizes the lateral movement of the aircraft.
Figure 142 shows a recording from a strip chart recorder when the rudder
is moved with (a) and without (b) the yaw damper. Recording of character-
istics w and 0) at low flight speeds was performed with flaps fully down.
After the rudder impulse was transmitted, the direction of the aircraft began
to slip with a bank.
As we can see from the recordings, after 8.8 sec k = 2 , after 12.1 sec,
1.94 and further, as the oscillations were damped, the value decreased.
Attenuation of oscillations shows the dynamic lateral stability of the
aircraft. The- value of k should lie between zero and one. We can see on
Figure 143 that this condition is observed at various altitudes only within a
definite range of M numbers, for example for H = 10,000 m at M > 0.75. At
smaller M numbers, k > 1. When the value of k is extremely high, so that the
ratio m /m is high, the aircraft will be judged unsatisfactory by its pilots.
This is explained by the fact that with high transverse stability, the reac-
tion of the aircraft to slipping becomes quite sharp. In this case, even
small slip angles cause the aircraft to bank sharply, and banking and yawing
movements with comparatively short repetition periods occur, and are not
always damped. This "rocking" of the aircraft is usually evaluated by pilots /225
as lateral instability, although actually it is an excess of lateral stabil-
ity, causing the aircraft to respond eagerly to the slightest random slipping.
In landing modes, the values of k produced are rather high (on the order of
of 1.5-2), leading to yawing and rocking of the aircraft (Figure 144).
Piloting of the aircraft is more difficult, and the pilot must frequently
operate the controls. Flight in bumpy air becomes particularly unpleasant.
x(V = 220 km/hr, 6
Determination of Value of
is the angle of devi-
ation of the rudder, H = 2000 m, landing gear
and flaps down)
The dependence of
the parameters T, k and
™bl ' characterizing
the lateral dynamic
stability of the
aircraft, on flight
speed are shown on
§20. Yaw Damper
We know that an
will have satisfactory
lateral stability if,
in addition to trans-
verse and directional
stability and the
optimal combination of
these two, it also has
good damping proper-
ties, providing intens-
ive damping of lateral
1 Ll 1 i
220 2(tff 2B0 2S0 V.XMjtir
Figure l43. Character-
istics of Lateral Dynamic
Stability As a Function
of M Number (angle x =
= 35°, landing gear and
Flaps Up); 1, 2, Normal-
ized values of parameters
Dynamic Stabi 1-
ity As Functions
of Fl ight Speed
( 1 . g . down ,
flaps down, H =
= 2100 m)
The installation o£ dampers has allowed improvement in the damping char-
acteristics in the event of perturbations to be achieved, particularly during
takeoff and landing. At the same time, the effectiveness of the ailerons has
Thus, the stability of an aircraft is increased and the work of the pilot
is greatly eased, especially in transient modes. For example, the yaw damper
provides automatic damping of aircraft course and bank oscillations by
artificially increasing the damping coefficient by automatically shifting the
rudder to an angle proportional to the angular velocity. As the yaw damper
operates, the intensity of damping of lateral oscillations is increased; this
means that the number of oscillations to complete damping and the total time
of damping are decreased. The amplitude of oscillations A (Figure 116) during
one period is decreased so greatly that the value m^ = A /A is decreased by
several times. Figure 142 b shows a diagram of the decrease in angular
velocities when the yaw damper is turned on after a pulse is fed to the
rudder. The period of oscillation is decreased to 5-7 sec, m, ^ = 5-8 and the
sense and significance of parameter k are lost.
The actuating mechanism of the damper (Figure 145) is a telescopic arm.
Control of the rudder during operation of the damper is performed using a
hydraulic amplifier which transmits the force to the rudder.
The angular velocity transducers, which measure w and w , are gyroscopes
with two degrees of freedom, reacting to the angular velocity of rotation of / 224
the aircraft about the oy and ox axes. As the aircraft oscillates about these
axes, periodic changes in angular velocities of yaw u and bank w occur.
Electrical signals are produced which are proportional at each moment to the
values of these velocities, then are amplified and sent to the telescoping
arms. The telescoping arms are installed in the arms of the rigid control
system from the pedals in front of the pilot. The hydraulic amplifier
deflects the rudder depending on the linear displacement of the shaft of the
telescoping arm according to an established control law. For example, with
the landing gear down and flaps down, deflection of the rudder occurs on the
basis of signals from the to and to transducers. The control law can be
^ y x
represented by the following formula:
where A6 is the deflection of the rudder;
A, B are the coefficients of proportionality corresponding to the
adjustment of the damper.
With the landing gear and flaps up, the signal from the co transducer is
disconnected and the operation of the damper follows the law A6 = Au .
The operation of the telescopic arms has no influence on the movement of
the pedals, although the rudder is deflected by an angle proportional to the
angular velocity of rotation of the aircraft. When the aircraft rotates to
the right, the rudder is deflected to the left and vice versa.
Let us use the following examples to analyze when and how the rudder is
deflected by the damper:
1. Let us assume that in flight with landing gear and flaps down, the
pilot turns to the right. To do this, he deflects the stick to the right,
banking the aircraft to the right by angle y (Figure 146 a) . Due to the
difference in lifting forces on the wings, transverse bank moment +M appears
from the ailerons, under the influence of which the aircraft begins to
rotate to the right at angular velocity +0) . As it banks to the right,
the aircraft will slip at angle +g to the right (lower) wing (Figure 146 b) ,
and lateral moments M and M appear.
tvvvwvvt=j ^ __ ' ^ ^ ~
Figure 1^5- Diagram of Operation of Yaw Damper in
Rudder System: 1, Pedal; 2, Spring load; 3, Trim-
ming mechanism; h. Telescopic arm; 5, Amplifying
unit; 6, Angular velocity transducer; 7, Hydraulic
amplifier; 8, Rudder
In a laterally stable aircraft, as slipping begins, transverse moment
M arises, acting to eliminate the bank, i.e., acting to lift the wing
(Figure 146 c) . This moment, proportional to the coefficient of transverse
stability m and slip angle g is:
-m gC (where C = qSt, q is the
velocity pressure, S is the area of the wing, I is the wing span) and acts
against the deflected ailerons, as a result of which the effectiveness of
transverse control is worsened. The greater the transverse static stability
of the aircraft (bank stability), which is a property of all swept wing
aircraft at low flight speeds (V = 240-280 km/hr) , the more sharply the
aircraft will react with reverse bank to the lifting (lagging) wing during
slipping, so that a position arises in which the ailerons are ineffective.
Due to the directional stability, as the aircraft slips to the right a moment
appears proportional to the coefficient of directional stability -M ^ =
= -m 3C, rotating the aircraft to the right at angular velocity -u /226
(Figure 146 d) in attempting to eliminate the slip, slightly reducing the loss
of effectiveness of the ailerons. Therefore, the less slipping at the moment
when the aircraft is banked, the less will be the bank in the direction of the
Thus, in order to increase the effectiveness of the ailerons, it is
necessary when the aircraft is banked to reinforce rotating moment M ,
adding a moment from the rudder resulting from its deflection by angle
+A6 . This deflection is created by the yaw damper.
With flaps and landing gear down, the deflection of the rudder from the
yaw damper is determined from the formula:
The signal m deflects the rudder by angle ^6 -. = Boi . However, due to
the appearance of the angular rotation velocity -to (rotation of the right due
to shifting of the rudder) the rudder will also be automatically deflected by
the damper in the opposite direction by angle -A6 „ = -Aw . The summary
deflection of the rudder +A& will be less than from the signal +« alone
(Figure 146 d) so that the effectiveness of operation of the damper will be
slightly reduced. However, the controllability of the aircraft (more
precisely, the effectiveness of the ailerons^ is increased significantly m
comparison to the controllability without this damper.
2. If the disruption of equilibrium of the aircraft occurs due to a gust
from the left (Figure 146 e) forming a right bank (we will consider that the
pilot has not yet had time to move the controls) , slipping onto the right
wing occurs at angle +g. As in the preceding case, lateral moments occur.
Transverse moment -M will bring the aircraft out of the bank, and rotating
moment -M will act to reduce the slip angle. Thus, as a result of the gust
we have +u and as a result of the slipping, -to . The rudder is deflected by
A6 , = Bo) in addition to A6 „ = -Ato .
rl . X r2 y
operation of the yaw
damper is more
complex than what we
have just analyzed.
In particular, after
equilibrium is dis-
moment -M results
in angular velocity
-0) (rotation to the
left) and the rudder
is shifted to the
left. However, the
action of angular
146. Explanation of Operation of Auto-
matic Rudder Control by Damper
created by action of
the pilot or a
vertical gust, since
the initial deflec-
tion of the rudder
the slipping. The
of the rudder may be
so great that the
aircraft reduces slipping onto the right wing energetically, even perhaps
beginning to slip onto the left.
In this case, a bank onto the right wing will appear again, and the
aircraft as a result will yaw back and forth several times, rocking from wing
to wing. The damper causes the oscillations to die out quickly, and the pilot
feels no sensible rocking.
Also in flight (flaps up, w signal disconnected) with momentary
application of a side wind gust, the aircraft will first energetically rotate,
and slipping occurs at angle 6. Due to the u signal, the rudder is deflected
by the damper to eliminate the slipping, and due to the action of the damper,
in addition to the damping properties of the aircraft, rotation under the
influence of the side wind will be retarded (for simplicity we will not
analyze the banking moment) . When, due to the directional stability and
deflection of the rudder to reduce slippage, the aircraft tries to return to
its initial position, w of opposite sign appears and the initial deflection
of the rudder is decreased. The effect of the directional stability of the
aircraft is slightly reduced. The movement of the aircraft will be directed
to eliminate the slipping, and it returns to its initial position, eliminating
the initial slipping, and may even begin slipping on the other wing. However,
these oscillations of the aircraft about the oy axis are rapidly damped and
rocking is eliminated.
The pilot may get the impression that the directional stability of the
aircraft with the yaw damper is worse, and that the aircraft is less stable,
although in actuality, the yaw damper causes perturbations which arise to be
quickly attenuated. Thus, each angular velocity of rotation of the aircraft
about the oy and ox axes corresponds to a definite deflection of the rudder.
If angular velocity u is 1 deg/sec, deflection of the rudder will be
Ao) degrees, while if u =1 deg/sec --6 = Bw degrees (A and B are equal
to about 1.5-2) .
In order to increase reliability of damper operation, usually two series
connected telescoping arms are installed, operating simultaneously. Their
control action is added. The stroke of each arm is 6-8 mm, and the maximum
deflection of the rudder by the damper is 5-6°.
When the rudder is turned off or when there is no angular velocity of
rotation of the aircraft, the telescoping arm automatically takes up a
The hydraulic amplifiers of the yaw dampers operate without reverse.
This means that the aerodynamic load arising in flight on the rudder is not
transmitted to the pedals, and the entire hinge moment from the rudder is
absorbed by the amplifier piston. The pilot need only expend the force
required to move its valve. Since this force does not give the pilot any
"control" feeling," the desired magnitude and nature of force change must be
created by inclusion of a special spring loading device in the control system.
When the pedals are moved (by the pilot) the load springs are compressed, /228
imitating the aerodynamic load from the rudder. The force from the pedal can
be removed (during long flight with deflected rudder) by an electromechanical
trimming mechanism which shifts the body of the spring loader to a position in
which the load is reduced to zero. In all cases of failure of the yaw damper,
control of the rudder is performed by the pilot with the pedals, requiring him
to overcome the hinge moment from aerodynamic loads.
§21. Transverse Controllability
Transverse control of the aircraft is performed by the ailerons, and in
certain aircraft by the ailerons together with interceptors. Deflection of
the interceptors (aiding the ailerons) is performed after the ailerons are
deflected by 8-10°. This type of control is characteristic for aircraft with
large wing areas. The effectiveness of transverse control of the aircraft is
Also, the ailerons are frequently made in sections, in order to reduce
"floating" in case of flow separation on the wing. The ailerons are usually
deflected by ±20° (up and down) , and the angle of rotation of the control
wheel is 120-180°. The angle of aileron deflection by the autopilot averages
±2.5-3.5°. In the portion of the wing where the ailerons are placed the
relative thickness of the wing profile is slight, 10-12%, the relative curv-
ature 0.8-1.5%. The comparatively small relative thickness and slight
curvature allows the ailerons to be deflected by the same angle up and down.
The rotating moment thus produced (as a result of difference in the drag of
the wings with ailerons up and down) is slight, even at large angles of attack
and has almost no influence on the behavior of the aircraft (rotation about
vertical axis) .
A swept wing shape has an unfavorable influence on transverse controll-
ability, particularly at large angles of attack. The tendency of swept wing
aircraft to react sharply by banking to slipping and to eliminate aircraft
banking (by operation of the ailerons) significantly decreases the effective-
ness of the ailerons. Their effectiveness is decreased by side flow of the
boundary layer along the length of the wing, increasing the intensity of flow
separation at its ends. Aerodynamic baffles prevent early development of flow
separation in the terminal cross sections and thereby increase the effective-
ness of aileron operation.
Let us look upon the force applied to the control wheel for ailerons in
order to create an angular banking velocity of 1 rad/sec, AP /Au as a char-
acteristic of transverse controllability, as well as the change in angular
0) resulting from a change in aileron deflection of one
degree, Atu /A6 .
During transverse rotation, a damping moment arises which should be
equalized by the banking moment from the ailerons.
0,5 0,6 aj 0,75 M
Force on Control Wheel As a
Function of M Number
As we can see
from Figure 147, at
M = 0.7-0.75, the force
is 105-156 kg. This
means that if we must
create an angular
0) = 3 deg/sec, a force
of 5.5-7 kg must be
applied to the wheel.
The higher u , the
is doubled, the force also doubles.
greater must be the
force on the wheel.
As the flight altitude is increased
with constant M number, the force on the wheel increases, since, due to the
decrease in velocity pressure, the aileron deflection angles increase. We can
see from the figure that at 10,000 m, the forces are greater than at
H = 6000 m.
a I I ■ T 1 I i I
The aileron effective-
ness can be estimated as a
function of M numbers and
altitudes using the graph on
Figure 148. The higher the
absolute value of Aoj /AS ,
the more effective are the
ailerons. At speeds
near the maximum
the effectiveness ot the
ailerons should allow the
development of an angular
velocity of cd =12 deg/sec,
with forces not over 35 kg
on the wheel (according to the technical conditions) . For example, at
H = 10,000 m and M = 0.75, the creation of to =1 rad/sec (57.3°) requires a
force of P = 156 kg at the wheel. If a force of 35 kg is applied, we produce
an angular velocity w =12.8 deg/sec. The aileron deflection used is
a,7 niB a^BM
Figure 1^8. Aileron Effectiveness As a
Function of M Number
The quantity 2.29 — |^- — — (0.04 ^, ^^ ) is taken from the graph of
The aileron effectiveness in a landing maneuver (M = 0.2, V = 250 kmA'i")
can also be estimated using the graph of Figure 148. As we can see, with an
aileron deflection of one degree we produce u =9.45 deg/sec (Aw /A6
= 0.0165) . x X a
With a force on the wheel of 90 kg at these speeds oj = 1 rad/sec, and
the production of an angular rotation velocity of 9.45 deg/sec requires a force
of 14.8 kg.
§22. Directional Controllability. Reverse Reaction for Banking
The rudder is deflected to the right and to the left by the pedals by
20-25°, by the auto pilot by an average of ±4-5°. Axial compensation of the
rudder is generally 28-29% of its area (in order to produce acceptable
forces). On most aircraft, it has been noted that, due to increased area of
axial compensation at angles of deflection of 10-12° or more (about one third
of the pedal travel) the tip of the rudder moves out into the stream and
forces on the pedal begin to decrease. A phenomenon of overcompensation
arises. In order to eliminate this phenomenon, the rudder control system
includes spring loaders. They compensate for the decrease in force on the
pedals at large deflection angles or during slipping.
Also, interceptors may be used. They have an angular profile and are
fastened to the front of the rudder in front of its rotation axis
(Figure 149) .
The action of an interceptor can be reduced to the following. When
the rudder is deflected by 10-12°, the interceptor on the left side enters the
stream and creates separation (and therefore a change in pressure distribu-
tion) in the portion of the rudder behind the axis of rotation. The inter-
ceptor on the right side is covered by the vertical tail surface and does not
interfere with the flow. Due to the rarefaction formed on the left side, the
rudder attempts to move to the left (move with the stream) , which creates an
additional load on the right pedal as the rudder is held in its deflected
position. As we can see from the graph, the force on the pedal increases with
increasing angle of deflection of the rudder, while where there is no
interceptor the force begins to decrease at deflection angles 10-11° (over-
compensation effect) .
Rudder to Right
Figure 149. Force on Pedals As a Function of
Deflection of Rudder During Straight Line
Flight with One Motor Off (V = 300 km/hr,
landing gear down, 63 = 20°, H = 1500-2000 m) :
1, Vertical tail surface; 2, Interceptor;
bank velocity; AtS is a change in rudder deflection
tion of the inter-
ceptor causes an
increase of the hinge
moment and produces a
direct force on the
pedals, this force
being greater, the
greater the angle of
inclination of the
Let us look upon
the banking reaction
of the aircraft to a
deflection of the
rudder defined by
Aw /AS as a char-
directional control -
ability, where Aw is
•^ ' X
the change in angular
of one degree.
As we can see from Figure 150, up to M
0.84-0.85, Ao) /A6 is positive,
i.e., the bank follows the control. At high M numbers, the sign becomes
negative, i.e., the bank is opposite. This means that a reverse bank reaction
occurs when pedal is fed. Let us analyze this feature of aircraft with swept
wings in more detail.
iSr, sec, deg
In a transversely stable aircraft
when left pedal is applied a slip to the
right occurs and, as a result, a moment
arises tilting the aircraft onto the
left wing; conversely, when right pedal
is fed, a bank to the right occurs.
This reaction of the aircraft to deflec-
tion of the rudder is called normal or
Aw /A6 on
= 10,000 m;
M Number (H =
at M = 0.84,
king reaction of
t to deflection
However, when an aircraft with
swept wings flies at high M number, this
regularity may be disrupted (for
example, when right pedal is fed, the
aircraft banks to the left rather than
the right) .
results from the
helps to increas
reinforce the di
is different at
The appearance of a reverse bank
reaction when the rudder is deflected
influence of compressibility of the air on the aerodynamic
of the wing. At subcritical speeds, the sweep of the wing
e the transverse stability of the aircraft and, consequently,
rect bank reaction to deflection of the rudder. The picture
supercritical flight speeds.
During slipping, the effective sweep angles of the right and left wings
change, so that their critical M numbers also change (Figure 151). The wing
which is moved forward shows a decrease in M as a result of the decrease in
effective sweep angle, while the lagging wing, on the other hand, shows an
increase in M as a result of the increased sweep angle. This change in M
means that in slipping the wave crisis develops at different times on each
wing -- first on the wing on which the effective sweep angle is less. This
time differential in development of the wave crisis on the left and right
wings and, consequently^ the asymmetry in the change of their lift, causes the
appearance of a reverse bank reaction when pedal is fed.
Figure 152 shows the regularity of deflection of ailerons during
acceleration with slipping in an aircraft with reverse bank reaction to
slipping. It is easy to determine the M number at which the degree of normal
reaction of the aircraft to slipping begins to decrease (point 1) and when the
normal reaction is transferred to a reverse reaction (point 2) . At this same
point 2, where the curve passes through zero, there is neither a direct nor a
reverse bank reaction to slipping. In other words, when flying with M number
corresponding to point 2 the aircraft does not have any bank reaction to
slipping; the manifestation of this is that when the pedals are deflected a
pure yaw motion occurs without any tendency to bank.
di recti on
riqht wing J [
Figure 151. Change in Effective Sweep
Anqle and Coefficient c As a Function
of M Number with Constant Angle a = 2°
for Wings Differing in Sweep Angle
•area of reverse
Figure 152. Deflection
of Ailerons During Accel-
eration with Slipping on
an Aircraft with Reverse
Bank Reaction to Slipping
Between points 2 and 3 we find the area of reverse bank reaction to
slipping. To the right of point 3, direct reaction is restored once again.
Frequently, this point is unattainable, since the corresponding M number is
beyond the limiting permissible number for the aircraft (as is the case on
Figure 152) .
The beginning of the reverse reaction can be found by accelerating and
deflecting the rudder.
If an aircraft with a swept wing (x = 35°) flies at a speed corresponding
to M (Figure 151) at which the reverse reaction occurs (M > M ), when right
pedal is fed during left slip, for example with an angle 3 = 10°, the
effective sweep angles of the wings change: the angle of the left wing is 25°,
of the right wing -- 45°. As a result of this, the development of the wave
crisis on the left wing is reinforced, while it is retarded on the right wing.
As a result, coefficient c on the left wing is sharply decreased, while it is
slightly increased on the right wing, leading to high transverse moments,
tending to bank the aircraft in the direction of the slip.
The greater the sweep of the wing and the thinner the wing profile, the
weaker the reverse bank reaction will be, since the change in c with M number
will be smoother. The M number corresponding to the point of intersection of /£M
curves c = f(M) for sweep angles 25 and 45° is represented by M . The pilot
should know the M number of the reverse reaction of his aircraft and recall
the factors which might lead to improper piloting if he is forced to fly at
M > M .
We note in conclusion that in modern aircraft the rudder is practically
never used in flight. Control of lateral aircraft movement (curves, turns,
spirals and other evolutions) are actually performed by the ailerons alone.
Exceptions include takeoff and landing, during which gusts of wind (particu-
larly side gusts) are sometimes countered using deflections of the rudder.
§ 23. Involuntary Banking ("Valezhka")
In high-speed aircraft with swept wings, so-called involuntary banking
may occur, which has come to be called "valezhka." This phenomenon occurs
both at low altitudes at high indicated speeds, and at high altitudes at high
M numbers .
Valezhka may occur for two reasons: a) as a result of the appearance of a
banking moment under the influence of a difference in lifting force on the left
and right wings and b) due to a drop in aileron effectiveness.
The difference in lifting force on the wings is created due to geometric
or rigidity asymmetry of the aircraft. Geometric asymmetry is characterized
by a difference in effective angles of attack of portions of the right and
left wings. If the wings have different structural rigidity and therefore
different deformations, a difference in angle of attack may occur. All of
this leads to large banking moments at high flight speeds.
However, this banking moment sometimes cannot be countered by deflecting
the ailerons, since under certain conditions their effectiveness is decreased.
Suppose, for example, a banking moment on the right wing appears. In order to
counter this moment, the pilot deflects the right aileron downward, the left
aileron upward. However, when the jiilerons are deflected at high indicated
speed (when the velocity pressure is great) moments appear which twist the
wing. Due to the elasticity of the wing, the angle of attack of the right
wing is decreased, that of the left wing increased. This diminishes the
effect of aileron deflection. The forces on the control wheel increase
sharply. This phenomenon is called aileron reverse.
At high altitudes, the aileron effectiveness drops due to the presence of
supersonic zones and compression drops on the wing.
In all -cases where valezhka occurs, the pilot should take measures to
prevent banking of the aircraft, and the bank should be corrected with the
ailerons. Countering of valezhka at high M numbers by feeding pedal against /235
the bank may result, in some aircraft with swept wings (as a result of the
reverse bank reaction) to an increase in the bank.
§2^. Influence of Compressibility of Air on Control Surface Effectiveness
The controllability of an aircraft, dependent on the operation of the
horizontal control surfaces, may change essentially at high M numbers. Let us
analyze the operation of the control surfaces at various M numbers. As we
know, when the surfaces are deflected at subcritical speeds, a change in the
flow spectrum and pressure distribution occurs throughout the entire profile
of the control surface, as a result of which aerodynamic force R, arises
(Figure 153 a) . The change in pressure distribution is explained by the fact
that deflection of the control surface creates small perturbations, propa-
gating in all directions at the speed of sound, including against the direc-
tion of flow, which is subsonic. These small perturbations cause changes in
pressure along the profile of the air foil.
Figure 153. Explanation of the Influence of Air
Compressibility on Control Surface Effectiveness
If flight is performed at supercritical M numbers, at which the wave
crisis is developed on the control surface, the effectiveness of the articu-
lated surfaces is decreased considerably. This occurs for the following
After supersonic velocities arise on the tail surfaces, when the pressure
jump ends, the deflection of the control surface can no longer change the
nature of the flow around the entire tail surface, nor can it change the
pressure distribution over the surface (Figure 153 b) . In this case, the
perturbations caused by deflection of the articulated control surface section,
propagating at the speed of sound, cannot extend to the portion of the tail
surface where the flow rate is higher than the speed of sound. Therefore, the
nature of the flow changes only over that section of the tail surface wTiich is
located behind the compression jump. Thus, the creation of additional aero-
dynamic force by deflection of the articulated surface includes only a portion
of the tail surface, so that the magnitude of the force is decreased.
In order to improve the effectiveness of the surfaces at high speeds,
for the tail surfaces can be increased by using high-speed profiles and
giving the tail surface an arrow-like form in cross section. In order to
prevent early loss of tail surface effectiveness, M should always be greater
for the tail surface than M for the wing.
Also, the horizontal tail surface
should be removed (upward or downward) from the vortex flow zone behind the
wing, in order to avoid decreases in its effectiveness.
§25. Methods of Decreasing Forces on Aircraft Control Levers
In order to control the aircraft, the pilot deflects the control surfaces
by applying certain forces to the command levers. The forces on the levers
depend on the hinge moments arising as the articulated surfaces are deflected.
If these forces are great and the flight requires a good deal of maneuvering,
operation of the control organs becomes fatiguing. At high speeds, signif-
icant hinge moments are characteristic, so that great forces must be expended
to control the aircraft.
ax I s
Mmj = o^o=^kS
Figure 15^- Explanation of Hinge Moment
and Operation of Axial Compensation (a),
and Diagram of Operation of Servo-
The hinge moment is
the moment created by the
aerodynamic force arising
rotation on the articulated
surface as it is
deflected relative to its
axis of rotation. This
moment acts against
deflection of the surface
and is perceived by the
pilot as a force on the
control stick or pedals
(Figure 154) . The hinge
moment increases with
increasing angle of
deflection of the surface
(from its equilibrium
position) , with the area
and cord of the surface
and with velocity
In order to decrease
the force on the stick,
axial or internal compensation, servo-compensators and trimmers are used.
Axial compensation is achieved by displacing the point of rotation of the
surface (hinge) backward, thus decreasing the hinge moment (Figure 154) .
Axial compensation of the elevator covers about 30% of its area, of the
rudder -- about 28-29% of its area, of the ailerons -- 28-31%. Greater values
of axial compensation may lead to overcompensation. Its essence is as
follows. The hinge moment can be decreased to zero, or if the hinge is moved
even further rearward a hinge moment of the "reverge" sign may appear. In
this case, the hinge moment appearing when the surface is deflected will tend
to increase the angle of deflection. This is an unfortunate phenomenon, and
is called overcompensation.
Figure 155. Internal Aerodynamic Compensation
(a) and Interceptors for Transverse Control on
Wings of DC-8 Aircraft (b)
On the TU-104
f _-J<;?v — ■ ~ aircraft, in order to
I h ^-"■-''''^^Ia^ . Servocompensator decrease loads on the
tion is used
(Figure 155) , which
is similar to axial
differs in that when
the control surface
compensation does not
extend beyond the
wing profile. Internal aerodynamic compensation is achieved by a plate
fastened to the front of the aileron. On one end of this plate there is a
sealing strip, the other end of which is fastened to the rear wall of the
nonmoving wing. This strip is a barrier, separating the internal cavity of
the rear portion of the wing into two nonconnected cavities. When, for
example, the aileron is deflected downward, the flow rate over the wing
increases, and the pressure correspondingly decreases. Due to the decrease
in pressure, air is pumped out of the upper cavity of the chamber and the
pressure in this cavity decreases. The pressure beneath the wing and in the
lower cavity increase. As a result of the pressure difference in the upper
and. lower cavities, aerodynamic force Y, acts on the strip and plate. This
force creates a moment about the axis of rotation of the aileron which
decreases the hinge moment. The compensation works similarly when the aileron
is deflected upward. The advantage of internal aerodynamic compensation is
that it produces a very slight increase in drag of the wing, since there are
no protruding parts of the aileron before its axis of rotation. However, it
does have certain defects as well. The aileron plates within the wing limit
the angle of deflection of the ailerons. For the elevator and rudder, which
have considerable deflection, the usage of this compensation is difficult due
to the thin tail surface profiles. The flexible strip must be carefully
maintained during operation. If the strip is damaged, the compensation
The servo-compensator (or Flettner) is a small supplementary control
surface located at the rear end of the main articulated surface and hinge
connected to the nonmoving portion of the tail surface (vertical tail surface
for the rudder or wing for the ailerons) by a tension member (Figure 154 b) .
Deflection of the control surface automatically causes the servo- compensator
to move in the opposite direction. The aerodynamic force arising on the
servo- compensator is opposite in its sign to the aerodynamic force on the
control surface. As a result of this, the hinge moment of the surface is
decreased. Servo-compensators are installed on the ailerons and rudder, less
frequently on the elevators. Servo-compensators are deflected by ±3-14°.
This reduces the force required to acceptable levels.
Trimmers allow loads operating over long periods of time and corre-
sponding to deflection of the rudder or aileron to be completely or almost
completely removed; they cannot be used to decrease the forces arising during
brief deflections of these surfaces (for example when moving into a new flight
regime or when countering external perturbation) .
The area of the elevator trimmer of a modern aircraft is 7-10% of the
area of the elevator, the area of the rudder trimmer is 8-10% the area of
the rudder, while the area of the aileron trimmer is 6-8% of the area of the
The angles of deflection of the trimmers are so selected that in case
of accidental operation of the electrical control mechanisms for the trimmers,
resulting in movement of the control surfaces, the pilot will be physically
able to hold the control surface in the required positions. For example, if
the trimmer of the rudder is deflected by ±3-4° and the rate of movement is
0.5 deg/sec, accidental operation of the trimmer will cause it to deflect
fully (in 6-7 sec) and at speeds of 300-350 km/hr, creates forces on the
pedals of 25-30 kg; at 500-600 km/hr at H = 1000 m, the force created is
70-80 kg. This force can be overcome by the pilot and copilot and represents
no emergency situation.
The angle of deflection of the aileron trimmers is also ±3-4°, and the
rate of movement is about 0.4 deg/sec. With the maximum deflection of the
trimmer, force on the control lever of 12-36 kg respectively is required for
speeds of 300-500 km/hr. The angle of deflection of the elevator trimmers is /239
6-8° upward, 8-10° downward, and the rate of movement is 1 deg/sec.
Accidental connection of the elevator trimmer electric drive and deflection of
the trimmer by 3-4° creates a load of 22-27 kg on the stick at 300 km/hr,
60-70 kg at 520 km/hr. Consequently, this also creates no emergency situation.
§26. Balancing of the Aircraft During Takeoff and Landing
Let us analyze how the aircraft is balanced during takeoff at 200-
300 km/hr (Figure 156) . At the moment when the front landing gear lifts
(V = 200 km/hr, takeoff with preliminary lift of front gear) , the angle of
deflection of the elevator 6 , = -16.7°, and the force on the stick is
37.5 kg. As the speed increases, the effectiveness of the elevator increases
and the pilot decreases its deflection, while the force increases. At the
moment of liftoff of the aircraft (V = 240 km/hr) , the angle of deflection of
the elevator is -14° and the force on the stick is 45 kg. After liftoff as
the flight speed increases, the elevator feed is decreased, and the force on
the stick also decreases.
- lift to H=25m
Usage o£ the
trimmer reduces the
force. For example, at
200 km/hr, a deflection
of the trimmer by one
degree decreases the
force by 3 kg, at
240 km/hr --by
4.35 kg, at
300 km/hr -- by 7 kg.
As we can see from the
graph, at 300 km/hr, in
order to remove the
force, the elevator
trimmer must be
deflected by approx-
imately 4°. Before
takeoff, the elevator
trimmer is preset at
1.5-2° (the wheel is
turned toward the
trimmer adjustment is performed in flight after the landing gear and flaps
have been raised.
Figure 156. Deflection of Elevator and Force
on Stick As a Function of Velocity During
H^ 15- 25m
Z70 260 250 2i0 230 22ir.\(>//ijt\r
Figure 157- Deflection of
Elevator and Force on Control
Stick As a Function of
Velocity During Landing
Balancing of the aircraft during
landing consists of the following. As
the velocity is decreased in the
glide, the deflection of the elevator
upward and force on the stick
increase. As we can see from
Figure 157, if the elevator is
deflected upward by 7 ° at 275 km/hr,
and the force is 28 kg (trimmer
neutral) , at 230 km/hr these
quantities are 13° and 38 kg
respectively. At the moment of
touchdown at 220 km/hr, the angle of
deflection of the elevator is approx-
imately 22° and the force on the stick
is about 60 kg. Actually, the trimmer
is used to decrease the force to
15-18 kg until the moment of touch-
down. Let us analyze how this occurs.
After the flaps are dropped to the
landing position (landing gear down) ,
the elevator trimmer is deflected by
7-10° to remove the force on the
stick. At 280-300 km/hr, the force on the stick is near zero. As the
velocity is decreased during the glide and the elevator deflection is
increased to 15-17°, the pulling forces on the stick increase, amounting to
10-15 kg at the moment of touchdown. An adjustable stabilizer allows the
loads on the elevator to be decreased significantly if it is deflected by
-2 to -5°.
Chapter XII. Influence of Icing on Flying Characteristics
§1. General Statements
In jet aircraft, icing generally occurs on the front edges of the wings,
vertical tail surface and stabilizer, the windshields of the pilot and
navigator, the temperature receptor and navigational instrument tubes
projecting outward from the fuselage and also the edges of the air intakes,
engine support pilons, blades of the intake directing apparatus and first /241
compressor stage. In modern turbojet aircraft with high power reserve, icing
of the fuselage, wings and horizontal tail surfaces changes the flying data
(flight speed, vertical velocity component, etc.) only slightly; the main
danger to flight under icing conditions does not result from an increase in
aircraft weight due to deposition of ice, but rather from the deterioration in
characteristics of stability and controllability of the aircraft.
The ice films which are formed (if the anti-icing system is not used) may
significantly change the wing profile and the profile of the horizontal tail
surface, creating increased turbulence and flow separation, which is particu-
larly dangerous for low speed flight during the approach to landing. Although
icing of the wings and fuselage change the flying characteristics but little,
icing of the stabilizer, even when the ice is rather thin, may have an
essential influence on the stability and controllability of the aircraft. Flow
separation on the horizontal tail surface depends primarily on the form of the
ice deposited and to a considerably lesser extent on its thickness.
Deposition of ice on the air intake, followed by separation of the ice
and entry of ice particles to the compressor blades may cause damage to the
compressor and to the engine. Therefore, icing of the intake channels and
first stage of the compressor cannot be permitted, not due to the decrease in
thrust which results, but rather due to the possibility of complete disruption
of compressor operation. Icing of the aircraft occurs primarily in clouds
(usually at temperatures below freezing), consisting of supercooled water
droplets which freeze when they strike the surface of the flying aircraft and
form ice deposits on various aircraft parts. The quantity of ice deposited
depends on the time which the aircraft spends under icing conditions. For
example, in flights of a TU-104 aircraft, icing was observed between 3000 and
8000 m at surrounding air temperatures from -8 to -34° in cirrus, alto
altocumulus and altostratus clouds. Icing has not been observed at high
altitudes outside the clouds.
The maximum time of continuous aircraft operation under intensive icing
conditions was 12-15 min, and the maximum ice thickness (according to the
indicator) was 46-50 mm. The brief time which the jet aircraft spends under
icing conditions results from the high flight speeds (650-850 km/hr) . Climbs
to 8000-11,000 m occur in 15-28 min, and the aircraft climbs through the
main layer of clouds near the earth (2000-4000 m) at high vertical speeds
(12-16 m/sec) in 3-5 min. The same thing occurs during the descent. The
greatest possibility of icing occurs during circling flight in the area of an
airfield, at which time the aircraft flies at 350-380 km/hr, spending
10-12 min in the approach to landing.
When flying at very high speeds, the surface of the aircraft is heated,-
which prevents icing to some extent. The surface of the wing is particularly
heated, since heat is liberated due to internal friction in the boundary layer
and the temperature of the leading edge of the wing is increased. There is a
point along the profile of the wing where the flow is completely decelerated,
which is accompanied by an increase in temperature AT of the air in relation
to the temperature of the surrounding air. This temperature increase depends
on the flight speed and can be calculated using the formula
where speed V is taken in m/sec.
The values of temperature increase for various flight speeds are shown in
However, during icing of an aircraft the actual increase is 30-50% less.
This results from the fact that the water droplets which deposit on the
surface of the aircraft will be partially o'r completely evaporated and
therefore will decrease the temperature of the surface. Also, heat exchange
occurs in the boundary surface, also reducing the temperature.
Types and Forms of Ice Deposition. Intensity of Icing
The forms of aircraft icing are various and depend primarily on the
extent of supercooling of the droplets in the clouds. The following types of
ice are differentiated'- :
^ 0. K. Trunov, ObZedeneniye Samotetov -i Sredstva Bor'by s Nimi [ I cing of
Aircraft and Methods of Its Control], Mashinostroyeniye Press, 1965.
a) Transparent ice (glaze) -- deposited on aircraft flying in medium with
large, supercooled droplets forming even, dense and transparent layer
(Figure 152 a). Ice formation temperature to -5°. This form of icing is
particularly dangerous, since it attaches itself firmly to the surface of the
aircraft. If there is a heating element on the front edge, barrier ice is
formed (Figure 158 e) ;
b) Translucent mixed ice -- encountered more frequently (Figure 158 b) ,
formed at -5 to -10°, sharply worsening aerodynamic quality of aircraft;
c) Hoar frost -- a white, large-grained crystalline ice, formation
temperature about -10° (Figure 158 c) , uneven deposition form with ragged
projecting edges, making flight dangerous (early flow separation possible);
d) Rime -- a white, fine crystalline deposit formed by water vapor
frozen upon contact with the cooled surface of the aircraft, representing no
danger for jet aircraft;
e) Barrier ice -- deposited on the leading edge at temperatures above 0° ,
on remaining portions at lower temperatures (the effect of the heating element
appears), the moisture which precipitates does not freeze, but is blown away
by the air and freezes to the surface of the wing (stabilizer) on both sides
of the leading edge, forming an ice deposit in a grooved shape along the
leading edge (Figure 158 e) . When deposited on the leading edge of the
stabilizer, may result in complete flow separation.
e) Heating element
Figure I58. Characteristic Forms of Ice
Deposits on Wings
stabilizer are also performed.
Since the testing of
an aircraft for stability
and controllability with
icing of the wings and
stabilizers represents a
particularly during the
warm season of the year, in
recent times tests have
been made using models in
wind tunnels with icing
imitators fastened to the
wings and stabilizer.
Flying tests of aircraft
with ice imitators glued
onto the front edge of the
As wind tunnel tests of model aircraft have shown, icing imitators
placed on the leading edge of the stabilizer cause slight changes in the
characteristics of stability and controllability. The forms of the imitators
(Figure 159) are similar to the natural forms of ice deposition. For example,
imitator form 1 represents the ice deposit produced during intensive icing
with poor operation of edge heater (the ice takes on the form of a groove) ;
2 represents barrier ice with the heating element operating; 3 represents the
deposition of ice at temperatures of -3 to -8° with the heating system not
The influence of icing of the stabilizer on characteristics of longitud-
inal stability and controllabiiity will be described below.
In order to estimate the degree of danger of icing of an aircraft, the
concept of the intensity of icing has been introduced, characterizing the
quantity of ice deposited (in mm) per min. The following scale has been
evolved: a) low intensity -- ice deposited at 1 mm/min; b) moderate -- from
1 to 2 mm/min and c) high -- from 2 mm/min up.
Figure 159. Forms of Imitators of Icing of Leading
Edge of Stabi 1 izer
§3- Influence of Icing on Stability and Controlab i 1 i ty of Aircraft in Pre-
landing Guide Regime
In order to estimate the influence of icing of the leading edge of wing
and stabilizer on the flying characteristics of an aircraft, as well as the
stability and controllability, special flying tests are performed under
conditions of moderate or slight icing at temperatures of the surrounding air
between -3 and -17°C between 1000 and 2000 m altitude with indicated speeds of
400-420 km/hr (speeds near those used in the landing approach) .
Piloting of an iced aircraft with an ice thickness of 30-40 mm on the
control surface profile (anti-icing system switched off) in horizontal flight
and during a climb with landing gear and flaps up without the creation of any
maneuvering loads does not differ :essentially from piloting under normal
conditions, i.e., with no icing. No noticeable changes in stability or
controllability of the aircraft were observed. The forces on the control
levers remain practically unchanged; no seizing or wedging of the elevator or
ailerons was noted. As the ice continued to increase in thickness, the motor
operating regime had to be increased by 4-5% in order to maintain steady
The data produced during wind tunnel testing of an aircraft model with
icing imitators on the leading edge of the stabilizer indicated that icing of
the leading edge of the stabilizer should not result in disruption of
stability or loss of control during sharp deflections of the elevator. This
allowed flying tests to be performed safely.
Sharp inputs of elevator control ("feed") during the approach to landing
at 260-290 km/hr (without icing) with landing gear, flaps and airbrake down
showed that the aircraft was stable in the longitudinal direction with overload
decreased down to 0.2. As we know, the pilot senses his control of the
aircraft from the resistance which he feels at the control stick during the /245
process of performance of various maneuvers. In order to create a consider-
able overload, large forces must be applied to the stick.
When the stick is "fed" forward, the pilot should feel a force on the
stick, greater the less the overload created. In those cases when the
pilot ceases to feel the control of the aircraft, longitudinal overload
stability of the aircraft is disrupted.
A reduction in the force on the control stick during icing conditions
results from a change in the hinge moments due to redistribution of pressures
on the horizontal tail surface. This is explained by the appearance of local
air flow separation over the lower surface of the stabilizer.
We can see from the graph on Figure 160 that at 290-260 km/hr as the
overloads decrease, the force on the control stick increases, as does the
angle of deflection of the elevator. The amount of elevator feed which must
be applied per unit of overload at 290 km/hr is less than at 250 km/hr. The
forces on the stick change as follows. For example, in order to create an
overload n = 0.4 at V = 260 km/hr, a force of 22 kg is required, while at
V = 290 km/hr -- 37 kg is required. With sharp deflections of the elevator,
the overload (particularly, 0.2) was retained for 3-4 sec and no drop in force
on the stick was observed.
Figure 160. Deflection of
Elevator and Force on
Control Stick As a Func-
tion of Overloads (pro-
duced in flying tests)
The model tests performed in the wind
tunnel using the horizontal tail surface
and negative angles of attack (-10 to
-18°) showed that when the leading edge of
the stabilizer is iced (the failure of
anti-icing system) , no disruption of
longitudinal static stability or change in
hinge moments of the elevator was
observed. A change in static stability or
hinge moment of the elevator is observed
only at angles of attack corresponding to
negative overloads. For zero overload
values, the graph m = f(a) in the case of
an iced leading edge of the stabilizer,
changes its inclination very slightly with
the three forms of imitators used, i.e.,
the longitudinal static stability remained
The flow angles were measured with
flaps down, and for wing angles of attack
of 2-4°, the flow angles were 5-6° (with
<f> = -2°).
As was stated above, when gliding in for a landing, the wing has a = 3°;
therefore, with a flow angle of about 5°, we produce a negative value of angle
of attack of the horizontal tail surface: a. ^ = a =d)-e = 3°-2°-5° =
-4° ^^ ^""^
With the same angle of attack, flow separation on a swept stabilizer
does not occur, since its critical angle of attack during icing changes from
16-17° by only 3-4° . Even with large flow angles (in the case of icing of
the leading edge of the stabilizer by barrier ice of considerable thickness),
the angle of attack of the horizontal surface does not change its critical
We analyzed the case in which tha anti-icing system did not work or was
not connected, and investigated what might occur if an aircraft began icing as
it descended for a landing. In practice, failure of the anti-icing system on
turbojet aircraft at V. , = 400-450 km/hr, the temperature drops along the
leading edge of the wings (hot air heating system turned on) decrease only
slightly, while the electrical stabilizer and vertical fin heating system
operate normally with one engine out, being independent of the number of
engines in operation on the aircraft. Greater difficulties can be created by
untimely switching on of the system heating wings, vertical tail surface
and stabilizer than by failure of one engine, with the resulting reduction in
hot air untake. It has been noted that when the anti-icing system on the wing
is turned on after ice has grown to 24 mm thickness on a controlled surface
the ice was shed from the heated leading edge in one minute, while when the
anti-icing system of the stabilizer was turned on, ice was shed from both
halves of the stabilizer in 1-2 cycles (2-4 min) . In order to be safe during
a landing approach with the anti-icing system not operating, the pilot should
bring his aircraft down smoothly, not creating overloads less than 1.
All related to a swept stablizer with x = kO-k5° '
NASA-Langley, 1969 1 F-5'<-2 241
JNational Aeronautics and Space Administration
Washington, D.C. 20546
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ATT t. LCU Bl)v■<MA^:, CHIEF, TECH. LIBRAky
If Undeliverable (Section
Postal Manual) Do Not R
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